The games listed here all support some sort of coop mode, except for most of the racing games. I've listed the racing games because, as with true coop, being able to race against humans and bots simultaneously is both fun and rare. I'll keep adding to this list as I discover/remember more coop games.
Serious Sam 1 & 2 - support Internet, LAN and up to four-way split-screen, cooperative multiplayer versus bots campaign mode. Allows coop game progress to be saved.
Quake 2 - LAN cooperative play versus bots (coop campaign). Progress can be saved. Watch out for friendly fire - it can't be disabled.
Unreal Tournament, UT 2003, UT2004 - a wide variety of LAN/Internet coop versus bot modes, allowing mix-and-match of humans and bots and adjustable bot skill levels and numbers. No campaign.
Battlefield 1942, Vietnam - LAN and Internet coop versus bot mode, allowing mix-and-match of humans and bots, adjustable bot still levels and numbers. No campaign.
Rise of Nations - highly flexible LAN/Internet coop versus bot modes. Bot players are individually adjustable in terms of skill, humans can team up with each other and/or with bots.
Dark Reign - highly flexible LAN/Internet coop versus bot modes. Bot players are individually adjustable in terms of skill, humans can team up with each other and/or with bots.
City of Heroes - server-based Internet gaming in which all human players are essentially on the same team, fighting computer-controlled bad guys. Fight them on your own or on teams or task forces. Team up with your friends or with total strangers on the net.
Diablo 1 and 2 - coop campaign vs. bots on LAN or Internet.
Whiplash - an old DOS racing game that supports two-way split-screen and LAN cooperative (team-based) gaming. I've never seen another racing game that does this.
Motocross Madness 2 - LAN and Internet competitive multiplayer with bots.
Need for Speed - LAN and Internet competitive multiplayer with bots.
TOCA Race Driver 2 - Split-screen, LAN and Internet competitive multiplayer with bots.
The entire EA sports NHL series - supports shared-screen, LAN and Internet coop gaming for up to five humans per team, with bots filling the roster on both teams. Game, season and dynasty progress can typically be saved.
Madden NFL - shared-screen coop versus bots.