Persona 4 Golden is currently the only game that has been tested and that won't run on the Steam Deck, aside from some VR titles.
When the Steam Deck was announced, Valve revealed that it planned to make as many games in Steam's library playable on the handheld console. It has since been verifying games on the platform and marking them as either, "Verified", "Playable", "Unsupported", or "Unknown". Games that are verified work well for the platform, while "Playable" means that they work with some limitations, such as text size.
According to SteamDB, Valve recently went through 67 games and marked their verification status, which isn't currently viewable through the store front itself but can be seen on SteamDB. Although most of the games were marked as either "Verified" or "Playable", there were a few that were noted as "Unsupported".
Although most of these were VR titles, which is probably no surprise, there is one title that sticks out among the rest - Persona 4 Golden. According to SteamDB, Persona 4 Golden is currently "Unsupported" on Steam Deck, which means that Valve has tried getting it to work on the handheld but has failed.
It's not clear what exactly would make Persona 4 Golden unplayable on a Steam Deck, as it's not the most graphically taxing game and doesn't have particularly small text. When you consider that games like God of War and The Witcher 3 have been confirmed to run on the console with no issue, it does raise an eyebrow as to what exactly is wrong with Persona 4 Golden.
For those worrying that they will never be able to play Persona 4 Golden on a console that isn't the PS Vita, first of all, fair, but it looks like the issue will be fixed in the future. The database entry says, "Valve is still working on adding support for this game on Steam Deck". This seems to imply that it will continue trying to add implementation, but it's a funny fate for a game that seems to be dodging console whenever it can.