LSD for the PS1. A dream simulator where you can aimlessly walk around fictional planes, complete with oddities only a dream would have. Single player only.
Most people claim it doesn't work on an emulator though, so you would be stuck with playing it on an actual PS1 or a PSP/PS3, and the game itself is really hard to come by if you don't download it from the Playstation Store or resort to perhaps less than legal ways of obtaining it.
Red Faction Guerrilla, a third person shooter with the main specialty being that you can destroy ANY construct you find. It has really smooth controls, a very wide assortment of weapons to both help you get rid of EDF (your enemies) and buildings. Single player is entirely free roam, in a pretty huge world too. When I ran across this demo, I played it for weeks being satisfied with simply destroying every building in that one mission with a 30 minute time limit. The multiplayer community is practically dead though. It's basically a forgotten pearl like Freelancer, where only small groups remain to play it. It uses a matchmaking system that sometimes takes 30 minutes to find a game, because of that, but if you can round up enough people, you can also play custom games.
Available for PS3, Xbox360 and PC. It's worth noting this game was made by Volition, and their other franchises (like Saints Row) are worth trying as well.