It depends on whether I've already been repeatedly smashing a specific key to get through the splash screens and intro and such. A mouse click usually works, but some of them require the escape button specifically.
However, I think the last game I played that had one of these screens was Assassin's Creed: Rogue, which I played with an Xbox controller, so I used the A button.
Enter key has been my go-to for as long as I have been playing games. At times the Enter key will not work and I have to use the mouse, E key, or some other fancy-schmancy combination. That is when my brain overloads and I start button smashing like a hippopotamus poked while eating.
Almost always space. Thanks to my typing teacher's otherwise mostly forgotten work (I was already typing up a storm way before learning the "right" way) I almost always have a thumb hovering over my spacebar, ready to pull the trigger and sometimes interrupt something with an unintentional jump.
As there's no real "Any" key, I always press "Space". It's probably my favorite key on the whole keyboard. It's big, so there's no chance of miss and thus it requires no effort to press. In my case "Space" is a full-fledged equivalent of "Any". Pressing something other feels unnatural...
Usually the games that make me "Press Any Key" are also console ports, which often means I'm using a controller, in which case I'll hit the Start button, since some of them also like to say "Hit Start" (even if on a PC that doesn't have a Start.)