Some RPGs like Fallout, definitely. I'd even run multiple games at once.
Action RPGs like Diablo; Absolutely. It's a requirement.
Others I've replayed were early Final Fantasy games. I've played Final Fantasy 3 (6) countless times. I've included challenges on myself like playing particular characters or without certain options (No Espers, No Grinding, Bad Character Groups [Umaro, Gau, Mog, Cyan]).
My favorite style play through is 100% completion. To trigger every event and get as much of your item inventory filled as possible. I think I got to 253/256 items. Missing Cursed Ring and Cursed Shield respectively and the last slot, I think, is meant to be empty. All while doing so in a low-level-game. No grinding. In fact, the lower your levels, the more speed you can put them on via Odin. Especially with Locke, every level saved adds to 1 point of speed. I once accomplished getting everyone with their levels being average 14 without save-states (console). (Gau was level 11, but I think that's partially because I used Leap and then left him there all game until later - lol)