Games can engage me almost indefinitely, as much as any other artistic medium.
When I was younger, I think I'd lean more to the gameplay side of things that could keep me thinking about the game every moment not playing it. Especially when it facilitated creative self-expression through build-craft, combat, strategy, etc. These days, I think I'm more interested in games that have evocative themes, stories, worlds etc., that give me something to think about when I'm not playing. Doesn't mean a game should not be fun though, I think the strongest and most engaging games are those that creatively utilize the interactive aspect of games to do something artistic. Nier Automata and Spec Ops: The Line are probably some of the most hardcore examples of that.
To also look the other way, the presence of real-world monetisation in games has a cancerous effect of engagement for me. Even when not implemented egregiously, money is the last thing I want to think about when trying to relax with a fun game or be intellectually stimulated by challenge, story, themes, etc. This is one of the main reasons I've not been super keen on Diablo 4. The story and visuals look awesome and just the kind of stuff I'd like. But it is ensnared in a financially incentivised never-ending 'live-service' format, which kills my excitement. Probably nowhere near as bad as I make it sound, but to me it sets it way back, especially with the breadth of cool games out there.