Sure, but not like you can on the other iPods (buying from the iTunes store). There are web sites that are made for the iPod Touch, and feature games and other applications (webapps). At the moment, you would need to hack your iPod Touch in order to actually have games on it permanantly.
You could also try this to save the webapps for on-the-go use, but it doesn't work all the time.
Or Your going to have to downgrade to 1.1.1 and go to jailbreakme.com on your iPod Touch and "Hack" it to get 3rd Party apps.
Just go to youtube and search "ipodtouchmaster" he has tutorials.
But please dont jailbreak its not worth it,it just wares down your ipod and makes it more unstable.But the worst that can happen is that you have to restore. if you have anymore questions just message me for my AIM