Defeating Morgana involves outdamaging her shield. If you have enough damage to burst through her shield (or ignore it), then you can kill her anytime. However, there are some caveats:
1. If breaking her shield makes you unable to kill her because of cooldowns, you'll never kill her without help.
2. If you can't repeat this damage consistently, she's just going to heal everything.
3. It's impossible as a melee.
So this leaves champions like Orianna, Urgot, Kog Maw, etc. All these guys can handle her.
You cannot ignore her and let her farm as Morgana is one of the strongest lategame mages in the entire game.
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By the way, you know how we're always dissing solo queue? Let me talk about an awesome solo queuer instead!
This guy, I thought he was some sort of super newbie with Maokai. He takes him jungle but we all stare at him while he doesn't place any saplings. So we're all like
What is this guy doing? Then blue spawns and he uses E. The ground pound. Now we're all like
Sigh... Randoms...
But it turned out this guy had a plan! An awesome plan!
He took blue and walked bot. Then comes the most incredible gank I have ever seen in my entire goddamn life. I shit you not okay. This guy. Level 2. He WALKS RIGHT NEXT to the enemy Caitlyn somehow (seriously, I'm still trying to figure out how he did this). No ghost. No flash. Then, 1 pixel next to her he finally roots her in place. Then the groundpound for even more CC and we manage to catch up and first blood this Caitlyn. Maokai. No saplings.
So cool!