Here's a quick list I put together...
XCOM is here. Safe download, no need to mess around with DOSBox, tested it myself.
The Spirit Engine 2, an indie RPG that recently turned free.
Final Fantasy 6: Needs no explanation
Chrono Trigger: Likewise
Terranigma: Accidentally included. While it does have one of the best stories I've seen in a game to date, it's also an action RPG. Sorry.
Rudra No Hihou/Treasure of the Rudras.
Nethack: You know this already.
Crawl: A rather odd roguelike that emphasizes skills, not grinding, not selling/hoarding, and is a bit simpler/fairer than Nethack. Still great. Very neat magic.
ADOM: Hates you. Hates you so much. Possibly the hardest roguelike around.
Incursion: A relatively merciful, very interesting D&D-based roguelike. Sadly partly unfinished.
Let me know if the games don't work/are boring/you don't know how to use emulators/you run out of games/whathaveyou.
P.S.: I also recommend Planescape: Torment and Arcanum, as well as the Etrian Odyssey series. That said, Arcanum is hard to find, and I can't afford it off GoG at the moment.
A few more recommendations: Etrian Odyssey for the DS, some rather good, if vicious dungeon crawlers with fantastic customization.
Age of Wonders 2: Wizard's Throne is a fun if poorly balanced TBS
Heroes Of Might And Magic 3 is fairly good too - I'd also be up for multiplayer of the former if you like.
Sorry about the wall of text, hope it helped. Get well soon!