Kuromukurohttps://myanimelist.net/anime/32245/KuromukuroLets start here since Jopax already mentioned it. It's a pretty good modern hybrid show made as PA Works 15th anniversary project and really needs more love. It's essentially a story of girl meets five hundred year old naked samurai and fights off an alien invasion. Some call it Inuyasha with giant robots but honestly I don't know anything about Inuyasha so I can't say anything on the topic. The characters are pretty fun and memorable, and it can be enjoyed by a more casual viewer since it mixes in a lot of hilarious slice of life downtime without detracting from the plot or tone of events.
Aim for the Top! Gunbusterhttps://myanimelist.net/anime/949/Top_wo_Nerae_GunbusterMade by Studio Gainax, and the first in their trio of great robot franchises, the other two being Evangelion and Gurren Lagann. It's a bizarre culmination of 80s scifi otaku nerdyness, and actually started out as a parody of Aim for the Ace and Top Gun of all things. It's a six episode OVA that starts out basically as described in the previous sentence following its main character Noriko and her training to become a top pilot, and eventually escalates into an over the top interstellar epic with a very famous and memorable ending. A lot of Gainax's staff that would go on to work on Eva also worked on this one.
Aim for the Top 2! Daibusterhttps://myanimelist.net/anime/1002/Top_wo_Nerae_2_DiebusterReelya mentioned this one before. It's a sequel/companion series to Gunbuster, and takes place 12000 years after the majority of its events. Tonally and aesthetically, its very different from Gunbuster, but at the same time has quite a few thematic connections if you analyze them together. Part of the reason for this was that it was made as an anniversary project for Gainax, and is in many ways a combination of all their previous major works, taking the setting of gunbuster, the coming of age themes of Eva and FLCL, along with some aesthetic cues from the former and a hint of the madness and disjointed scene jumping of the latter, along with various other plot elements and themes from their other works such as Mahoromantic. It's opening is also very famous and catchy. Can be considered a precursor to Gurren Lagann in some ways.
Gundam 0079https://myanimelist.net/anime/80/Mobile_Suit_GundamYou can't really have a mecha list without Gundam, so prepare for quite a few entries here. This is the original. The ol great one by Yoshiyuki Tomino from 1979. Gundam is the series that created the Real Robot genre as a whole, and its pretty much impossible to not have encountered it in the anime fandom due to its sheer bulk and influence. The original Mobile Suit Gundam was a war drama in a time when super-heroics and fantastic adventures were in vogue, and remains a classic story who had big effects upon the anime industry as a whole. Its animation is spotty even for its time, but the characters and story are incredibly memorable with themes that resonate to this day. It's also the start of the Gundam franchise's mainline Universal Century timeline, and having some knowledge of its events is pretty much required if you want to get into many of the bigger entries in the franchises' classic entries. For those intimidated by its length and datedness, there is a movie trilogy compiling the series that can be watched as a shortcut, but it should be noted many events are cut or changed to accommodate the runtime, and the pacing can become exhausting for a new viewer if they try to watch multiple movies in a single sitting. Note that all other entries I mention here do note require watching this to understand them however, and if you want to get into the ones that do then that's a whole other story.
Gundam 0080 War In the Pockethttps://myanimelist.net/anime/82/Mobile_Suit_Gundam_0080__War_in_the_PocketOne of many side stories in the Gundam franchise that take place simultaneously with the original, War in the Pocket is a six episode OVA that is considered one of, if not the, greatest stories in the franchise. Its a very strong and concise condensation of the themes of the franchise as a whole, and is incredibly heartfelt and memorable. Voted by anime watching adults in Japan to be one of the top animes likely to make you cry.
Gundam 08th MS Teamhttps://myanimelist.net/anime/81/Mobile_Suit_Gundam__The_08th_MS_TeamAnother side story in the original timeline, 08th MS team meanwhile is something to look into if you want a more hardcore and down to earth military story. Basically a Vietnam War movie with giant robots, dealing with the exploits of the titular 08th MS team on their deployment in southeast asia, and the burgeoning romance between its main character Shiro Amada and the enemy pilot Aina Sahalin.
G Gundamhttps://myanimelist.net/anime/96/Mobile_Fighter_G_Gundam?q=g%20gunAnd now for something completely different, G Gundam was the first AU gundam series made by Sunrise to bring in new viewers without being bogged down by the historical weight of the Universal Century timeline. Unlike other Gundam series, G Gundam is a super robot series that is a wuxia martial arts story in place of a war drama. It is essentially about an enormous tournament between nations on an environmentally ravaged earth where representatives of each space colony nation battle in out with national stereotype filled martial arts battles to determine who will control the planet for the next few years. Utterly memetastic and the origin of the SHINING FIIIIIINGER
Gundam 00https://myanimelist.net/anime/2581/Mobile_Suit_Gundam_00?q=gundamMy personal favorite of the alt timeline gundams, Gundam 00 is what I personally consider the best of the post 2000 Gundam AU series, despite its controversial second half. A pretty decent starting point to get into the franchise if you don't want to dive straight into the older shows. Unlike other Gundam series, it takes place on a near future earth that is politically and economically an extrapolation of modern geopolitics, and touches on several issues of current events such as war in the middle east. Plotwise its something like Isaac Asimov's foundation series, but with a terrorist organization in place of an experimental political state. Also something of a biblical allegory when analyzed enough. It's a bit... chuuni, but its all part of the fun and does a good job of conveying the aspects of the franchise's themes that it chooses to focus on provided you don't take it too literally. Also at the time it was made, Sunrise was running high off of the money it gained from Code Geass and Gundam SEED Destiny, and so the animation is consistently gorgeous for almost its entire run.
Eureka Sevenhttps://myanimelist.net/anime/237/Eureka_SevenHonestly I can't say much on this one because I haven't watched it yet, but its pretty much required to be here. Very famous for its central love story, and well loved by many. Its sequel and movie spinoffs on the other hand not so much.
Giant Robo the Animation: The Day The Earth Stood Stillhttps://myanimelist.net/anime/2321/Giant_Robo_the_Animation__Chikyuu_ga_Seishi_Suru_HiMade by the same director as G Gundam and so has a lot of the same tropes. Has a very interesting story of how it was made, because essentially the studio was able to gain rights to adapt the Giant Robo series as an OVA, but could not gain the rights to any of its side characters because of a competing production that was occurring at the time. So instead, they made their own plot and populated it with the casts of every other manga that was made by its original author, resulting in a martial arts filled dieselpunk scifi story featuring detectives, ancient chinese martial artists, ninjas, figures of the Three Kingdom Era, giant robots, magical girls, psychics, and all sorts of weird and awesome stuff all mushed into a single masterpiece of a story.
Space Runaway Ideonhttps://myanimelist.net/anime/930/Densetsu_Kyojin_IdeonOne of the main inspirations for Evangelion, Ideon was made by the creator of Gundam at the height of his depression issues. It essentially starts as a super robot story in space about a group of people who dig up an ancient robot and are pursued by an alien race that wants its power, but eventually spirals downwards until its eventual ending covered in the Ideon Be Invoked movie.
Tengen Toppa Gurren Lagann
https://myanimelist.net/anime/2001/Tengen_Toppa_Gurren_Lagann
The other, other, big Gainax mecha series. An enormous homage to Getter Robo, initially dealing with adventures of Simon and Kamina, two boys who live underground only to discover the surface world is oppressed by an army of beast people, and begin a rebellion to free humanity and bring the sun back to them. Famous for its enormous escalation in power-levels and crazy-awesomeness, and overall incredibly fun and sometimes wacky, but also somethings heartwrenching.
The Five Star Storieshttps://myanimelist.net/anime/2042/Five_Star_Stories?q=five%20star%20storieshttps://myanimelist.net/manga/1659/Five_Star_Stories?q=five%20star%20storiesBest described as Dune with giant robots. Or a version of Warhammer 40K if it had been based on shoujo, mecha anime, and hair metal instead of dark scifi and death metal if you're more familiar with that I suppose. It's hard to explain, but basically is about an incredibly high tech society ruled by a noble class descended from the genetically engineered mech piloting super soldiers of a long gone civilization and centers around the story of the God Emperor Amaterasu no Mikado, also known by his human guise Ladios Sopp. The manga is hard to find and the translation is unfinished, but I would highly recommend it regardless. Has an OVA movie adapting the first volume, and its just incredibly pretty.
Super Dimension Fortress Macrosshttps://myanimelist.net/anime/1088/MacrossThe other major real robot franchise. Macross is famous for its transforming plane robots, music, and love triangles. Macross is better known in the states as the first third of Robotech, following the crew of the SDF-1 and their battle with a race of giants called the Zentraedi. It was originally conceived as a parody of Gundam but shifted gears into being a serious story early in production, resulting in a very unique tone whose oscillation between serious and wackyness somehow manages to balance out into hitting a sweetpoint where its weirdness winds up making it seem actually closer to reality than not. Also notably has a whole stretch of episodes dealing with the rebuilding after the big conflict, instead of just ending it at the battles end. Reelya already covered Macross Plus and Macross 7, but there's also Macross Frontier, which is a more modern entry that was quite well liked at its time. However there are a few plot points of it that might make as much sense if you're not familiar with this series.
Full Metal Panichttps://myanimelist.net/anime/71/Full_Metal_Panic?q=full%20metal%20panicA strange blend of romcom and hardcore tactical combat and espionage. The first season is kinda meh, but its third and fourth seasons Second Raid and Invisible Victory are absolutely incredible. The second season Fumoffu is more of a comedy spinoff and is not required watching, but its an absolute treasure. Probably the only series I've watched where the idea of the main hero and heroine being from "different worlds" is actually visibly tangible.
Armored Trooper Votomshttps://myanimelist.net/anime/2582/Soukou_Kihei_Votoms?q=votomsHaven't seen this one, but worth mentioning because its basically the most down to earth of real robot series where the mechs are essentially disposable. Apparently its plot is space oddesey-esque.
GaoGaiGar King of Braveshttps://myanimelist.net/anime/890/Yuusha-Ou_GaoGaiGar?q=gaogaigarThe most famous of the 90s Brave series, basically THE 90s super robot franchise that constantly gets referenced in modern anime because many of the staff members now grew up watching these shows. Known for its hotbloodedness and escalation of tech and screaming. The Brave series is notable for their usage of sentient mecha similar to transformers, even actually using some leftover assets from the transformers franchise as designs in the series.
The Vision of Escaflownehttps://myanimelist.net/anime/182/Tenkuu_no_EscaflowneA very, very 90s show. Sortof a mecha, sortof a fantasy, sortof a romance. Its about a girl who finds herself brought to a fantasy world named Gaia and uses her fortunetelling skills to help in the fight against the Zaibach empire. More well known in the states than in japan thanks to its Fox Kids dub, yet surprisingly influential nonetheless.
Magic Knight Rayearthhttps://myanimelist.net/anime/435/Magic_Knight_RayearthActually a magical girl series by CLAMP, but mecha start showing up at the halfway point and become increasingly relevant from then on. Hard to talk about the plot due to some very major twists, but in simplest terms is about three girls who are summoned to the magic land of cephiro to fullfill a prophecy by rescuing the nation's princess from her captors. Very pure and cute, but also badass when it wants to be.
Code Geasshttps://myanimelist.net/anime/1575/Code_Geass__Hangyaku_no_LelouchJIBUUUNNNNN WOOOOOOOOOOOO.
A combination of Gundam and Death Note, taking place in a world where the american revolution failed and some weird stuff happened with napoleon, resulting in the northern americas becoming the nation of Brittania, who years prior to the start of the show conquered japan and subjugated its people, renaming it area 11. It follows the disgraced prince Lelouch Vi Britannia who dreams of revenge against his family, and gains a power called Geass that lets him give any people a single order they are forced to obey. He uses this power to become the leader of the japenese rebellion against Brittania in order to fullfill his goals. Known for lots of wham moments in the plot and cool fight scenes, with lots of intrigue between Lelouch and the Brittanian royal family.
Genesis of Aquarionhttps://myanimelist.net/anime/478/Sousei_no_AquarionA wacky super robot throwback by Shoji Kawamori. A strange story that in some ways is a parody of Evangelion. Only exceeded in its amount of mecha piloting related innuendo by Darling in the FRANXX. Basically is about super powered kids and their relationships with eachother as they deal with the Shadow Angels which are attacking humanity, along with their connections to the events 12000 years ago where the shadow angels were sealed away by their preincarnations. Has a sequel called Aquarion EVOL which is kinda famous for horribly crashing in its second half, and also a spinoff called Aquarion Logos that is basically the exact opposite in that it starts of weak but grows incredibly strong as it goes on further.
The Big Ohttps://myanimelist.net/anime/567/The_Big_OLike Batman with giant robots. You mentioned seeing the abridged version so I'll be brief.
Idolmaster Xenoglossiahttps://myanimelist.net/anime/1694/iDOLMSTER_XenoglossiaNormally wouldn't go on a recommendation list but I have a soft spot for it so I'm putting it here. And hey, you said you wanted weird after all. Basically when you ask the studio that makes most giant robot shows to make an Idolmaster show without giving them any direction about what kind of show it should be. Features alternate versions of the Idolmaster girls reimagined in a giant robot series, with all the changes to them that would be required for such. Surprisingly good once things start getting in gear, and once shit hits the fan it goes places you really would not expect given the source material. Features what is memetically known by fans as the greatest love story of all time.
Shinkalionhttps://myanimelist.net/anime/34985/Shinkansen_Henkei_Robo_Shinkalion?q=shinkalionPersonally I'm putting this here more out of self indulgence, but this is a very fun show airing right now that's basically the revival of 90s style super robot series. It's a kid show about transforming train robots, but a very well done one. Reportedly sales and ratings for this show are doing incredibly well in japan, so we could see a larger revival springing from this should trends continue. Notably the first few arcs are spent with the adults trying to find alternatives to having kids pilot the giant robots, and the loving relationship between the main character and his father stands out a ton in a genre full of memetically horrible dads. I could go on about that if you want but I'll be be here since that deserves its own post. Also features Hatsune Miku as a reoccurring cast member and has a crossover episode with Evangelion.
Not joking.Brain Powerdhttps://myanimelist.net/anime/1542/Brain_Powerd?q=brain%20powerdConsidered one of the "Eva Clones" by some, but was actually developed in tandem with it. A very bizarre post apocalyptic story where the mechs are organic beings spawned by an ancient ship called Orphan, and are treated like the adopted children of their pilots. Deals a lot with parent relationships. This is one of the shows where Yoshiyuki Tomino is at his most bizarre, and so making sense of it requires a lot of paying attention and inferring from what details you're presented with. Not for everyone, but I enjoyed it.
Heromanhttps://myanimelist.net/anime/4334/Heroman?q=heromanA lightheared story of a boy and his robot man fighting aliens. Made in collaboration with Stan Lee and possibly the most american anime ever
Break Bladehttps://myanimelist.net/anime/22433/Break_Blade?q=break%20bladeTakes place in a fantasy world and is about the one guy in it who can't use magic, which in this world allows people to control quartz and is used to make various forms of technology like the giant robots in it. However it turns out this trait allows him to pilot the excavated ancient machine known as the Delphine, which is a conventional mecha in the middle of this world of magic. This however gives it several advantages due to being made of metal instead of quartz, and this is used in several imaginative ways. Is a manga which was adapted into an OVA, which then later got recut into a 13 episode show with extra footage added in.
Mazinger Z
The Great Grandaddy of all piloted mecha. The classic story of Kouji Kabuto who receives a giant robot from his grandfather to fight the monsters of Doctor Hell. Can't really reccomend the original series because of its dated format and animation, but its worth checking out the modern spinoffs like Mazinger Infinity, Shin Mazinger, Mazinkaiser, Mazinkaiser SKL, and the manga Shin Mazinger Zero.
Getter Robohttps://myanimelist.net/manga/1551/Getter_Robo?suggestionI'm actually recommending the manga here, not the anime, but if you wanna check out an anime version of this start with Getter Robo Armageddon, it's an OVA who's plot really doesn't make much sense, but portrays the tone and madness of the manga very well.
The Getter Robo Saga was the primary inspiration for Gurren Lagann, which could be said to be a toned down version of Getter with its cosmic horror elements revoked. It begins with the Getter Robo and Getter Robo G manga, which are pretty simplistic and generic super robots save for their off the walls ultra violence, but these are short and the real gem is Getter Robo Go, which takes place 20 years later and blurs the line between Super Robot and Real Robot until it's almost invisible, and begins the process of escalating the story into a grand cosmic epic about humanity's destiny and the nature of the universe. After Go, read Shin Getter Robo, followed by Getter Robo Arc (also known as Getter Robo Ah, depending on whether you write the title in phonetics or in the intended sanskrit).