Any game where an average encounter will pit you against dozens or hundreds, maybe even thousands of enemies and expect you to win. Most of the time, it's really not that bad and makes you feel spectacularly badass.
Keep it to games where this is normal most of the time, where fights against fewer enemies is unusual. Try to avoid games that only have one or two zerg rushes or "war sequences" as TV Tropes calls them.
Koei Warriors franchises
Probably the quintessential "one versus one thousand" games. You take a historical figure (or an anime or Zelda character), you kill implausible numbers of rank and file soldiers, often with equally implausible choices of weapons such as giant paint brushes, kendamas, or nunchaku, and feel like an immense badass doing it. These games are power fantasies incarnate, and few things feel quite as satisfying as unleashing a rage attack in a crowded area and killing hundreds of unfortunate bastards whose greatest flaw was wearing the wrong color uniform.
Dead Rising series
If you prefer your hordes to be more the shambling, undead type, this is your deal. Whether you're trying to score the big scoop as Frank motherfuckin' West, get your daughter to safety as Chuck Greene or…doing whatever you do in the third game I haven't played, you can expect hundreds of rotting stereotypical 'murcans to be in your way. Being able to beat/carve up brain munchers with anything you can get your hands on is half the fun, and it's precisely because of Off The Record's time limitless sandbox mode that I consider it the strongest game in the series thus far.
Serious Sam
Hearkening back to a time when cover was just in the way, Serious Sam throws you into hordes of aliens and monsters that will start attacking your place en masse. From alien skeleton horse-things to rocket launching biomechs, chaingun-wielding scorpion people to the iconic screaming kamikazes, you'll be thrown mindless mook after mindless mook to mow down with bullet, rocket, laser, or cannonball. Hope you've practiced circle-strafing and dropped some cash on a good sound system!
EDIT: I can't, in good faith, post about the glory of Serious Sam and not include the wonderful dulcet tones of the Headless Kamikaze. I have rectified that.