They way I see it, genre-mixing in gaming has almost always lead to positive results. The go-to example is Persona, obviously, with its social simulating day sections and dungeon crawling night sections, all tied to together with a plot that somehow manages to make these two features seem natural together. Another well less known example, but an equally well done one in my opinion, is Puyo Puyo Box, mixing the classic JRPG formula and falling blocks puzzle games. I've always found these kinds of games interesting, and I'd love to hear about more games that do this genre mixing.
If you like Persona, you might like Catherine, same studio. It's kinda hard to describe, there's puzzle-block elements and date-sim elements and ??? Presentation is FLAWLESS tho. Really nice voice acting too. The versus mode is played at EVO, which will make no sense when you first see it, but it's surprisingly deep. Aside from that, it depends what genres you want mixed and how thoroughly, but there's almost a game for every person at this point. I just played Furi, which is a boss rush game (like Punch-Out! or Shadow of the Colossus) that combines twin-stick shooter mechanics with character action combat. Overwatch is basically a TF2 moba. I could go on, but if you're looking for recs...
Just about every shooter nowadays uses RPG mechanics now. Got levels for this and that, stat increases, customization, whathaveyou. If you want to look for something a little more in-depth than that though, there's still plenty of other games that have other genres within them. Pretty much any minigame in those big openworld RPGs can be considered genre-mixing. The thing is, sometimes games try to add too much and end up diluting the main focus or mechanic of the game leading to something less...desirable.
The problem I have with these cases is that more often than not the mini-game or extra mechanic is poorly implemented, or simply just broken completely. While its hard to call it an "extra mechanic", any DS game that uses even slight microphone input throws me to a complete dead end. Thankfully, this isn't used too much, but when it is, like in Spirit Tracks, it takes me highly unreasonable amounts of time to move on with what is essentially blowing into a microphone. Then there's a level in the DS port of Plants vs. Zombies that requires the microphone to constantly be stimulating to stop your units from falling asleep. I don't have the lung capacity for that, and I doubt anyone really has. Now that I think about it, I suppose you could consider just about any "rogue-lite" genre-mixing. Very few games are actually "true" rogue-likes, such as Rogue and Nethack, but mix the general concept with other genres; Isaac, Downwell, Nuclear Throne, Teleglitch, Rouge Legacy, Selunky, FTL... the list is quite vast.
