The first computer I used was a 286 XT (or AT?), and it had a turbo. I remember playing a few games on it where the turbo button came in handy.
One specifically that I remember was called freefall, and simulated diving into a cavern. You had to move the ascii character representing yourself back and forth to avoid the cavern walls as you fell. Simple, but fun.
Elite was very hard to play on some machines if they didn't have a turbo button. (To turn off turbo, and put the machine in a nice 12 MHz.)
Later machines kept the button, but either just hooked it up to a display or even left it disconnected. Cases could fit different motherboards so it's handy to have the button for the older motherboards.
When I was very young, we had a computer which had this button. I always wondered why pressing it would change HI to LO, as the name of the button suggested otherwise. I never really used it as my few games worked fine without it. I was too young to remember the name of the computer, and we got rid of it long ago.
I opened up my PC (486dx) and found that the Turbo button had been disconnected, and (obviously) made absolutely no difference whether it was pressed or not. I never really investigated further.
Some games didn't use a timing loop, they were written with timing that corresponded directly to CPU speed. Thus a turbo mode would make the games impossibly fast to play.
Sopwith and older versions of Scorched Earth come to mind as examples of this phenomenon.
I think this would be better as a poll rather than a story, there's not a lot to say.