It's difficult to say really, as our esteemed colleague Prometheus notes; most games really do feel as though they were made for this thing. So here are some of my favourites that I'm playing right now, and which emus they're on:
PCSXreARMed (we need to shorten this name for referencing!):
Dark forces - works like a charm, and it's bloody good fun.
Front Mission 3 - Tactical mech combat, and it's flippin' great.
Parappa the rappa - as long as you can remember what the PSX button layouts are. Still make me laugh 'kick, punch, it's all in the mind!'
Snes:
Earthbound - This is such an off the wall game, the humour is a really refreshing and it really is a nice break from all of those ridiculous cookie cutter jrpgs (seriously, one more 17 year old protagonist with spikey hair and I'm going to turn into a foul smelling cloud of steam), and it's really excellent to play on the Pando.
UAE:
Indy Heat - The very first game that I tried when I got my unit, and I had to complete it. I don't care what anyone else says, Indy Heat is one of my most beloved games. I really have a soft spot for isometric racing.
Captive - you'll need a mouse to enjoy it properly imho, and probably a quick nose through the manual, but it's such a cool game.
Native:
Streets of rage remake - as others have said, it's an extremely worthy addition to the SOR universe.
Beat of rage - Plays very well, and some of the mods work too, or you can make your own if you're that way inclined.
Heretic/Hexen - Once you've got the controls set up to your liking they're deliriously good fun.