Since games like Path of Exile and Planetside 2 have been suggested, I'm going to mention Marvel Heroes 2105. It's a fairly casual ARPG with some MMO elements (but not a full blown MMO). I'm not sure what the general consensus is on it as a good/bad game around here, although it seems to do quite well in reviews, player and otherwise. But within the context of this thread, I think it qualifies as not being Pay 2 Enjoy.
The main things you can buy with cash are unlocking new heroes to play (there's about 40 available), alternative costumes/looks for your heroes, boosts, and stash space.
New heroes can be earned through a resource that drops from killing any enemy - time to unlock depends whether you go for random heroes (cheapest) or specific heroes (price varies), but going the random route I've been unlocking them faster than I can play them to level cap. They also recently altered the newbie experience to let everyone play the ~10 starter heroes up to level 10, along with enough in-game resource to buy 1-2 heroes from the 2nd day they log in.
Costumes seem to be somewhat harder to unlock through play, but I've not actually tried to unlock one so I'm not sure. Seems to involve a lot of farming and luck though.
Boosts (xp, rare item find, special item find) can be gained semi-randomly via a resource earned running 'terminal' missions, a handful from specific story missions, and the devs frequently release promotional codes that give free boosts (most of which they'll email you if you're on the mailing list). There's also pretty much always one event or another running, some of which give server-wide boosts.
Stash space is the only thing from the cash shop that you can't earn in game. The free allowance of stash space is pretty limited though - you definitely feel the hurt there if you play beyond the short term (it's not essential to enjoy the game, but with all the crafting components, boosts, and gear for all the different heroes dropping, there's a lot of scope for hoarding and very little default space to hoard it in).