Weighing Your Options
iPod Touch games are downloaded either initially to your computer and then synced to your iPod Touch or downloaded through the App Store app on your iPod Touch and then transferred over to your computer to save. You obtain the purchased or free games from a single account and then have the ability to download them again. The primary storage space for these apps is not on the iPod Touch itself in most cases, but instead on the home computer. In this way, you have to recognize that they are not just software installed onto an iPod Touch and then kept there, as they are tied into a store account. The person who purchased and downloaded those apps has the rights to them.
The question about “if I buy an iPod Touch can I keep the games," can only be answered by the purchase agreement. If you buy a refurbished iPod Touch, this is not possible at all, as it has been returned to its original position and is like new. It does not have any non-original content installed, meaning nothing that was downloaded and added to it.
If you purchase the iPod Touch from a third party, such as on eBay or in a personal agreement, there may be content on it. The problem here is that the content will then be tied to the previous accounts, especially the Game Center log in. The easiest way to customize the iPod is to restore it, returning it to the factory settings. This would then remove all of the iPod Touch games and apps, though you would be able to tie it in to your Apple ID and iTunes account. If they choose to just give you their iPod Touch without any changes made you could, in theory, use all the games that are on that iPod. You may then be able to transfer them over to your computer, giving you the ability to re-sync them. This is not going to work as a guarantee, and if you will not be able to then update them. Since the user will likely want to use their iTunes account and will make you use your own on your newly purchased iPod you will then have your download history. This will cause some issues when there are games there that were not purchased on your account, but this will likely not be a problem for free games.