Ys I&II are very fun, quirky action RPGs that've finally been translated to English by fans; instructions on how to get them and patch them here ->
http://www.the-underdogs.org/forum/v...ic.php?t=43173
I suppose pretty much any Japanese RPG would count as an anime game, from the Final Fantasy series, Chrono Trigger, Secret of Mana etc.
Then there's this:
http://www.the-underdogs.org/theme.php?id=2
[edit] Ooh, ooh! Cave Story! <3