I don't think it's referring specifically to anyone. It could easily be the Toymaker, although The Celestial Toymaker never does anything to suggest he's mad.

I think kidnapping people, forcing them to play games, and turning them into puppets if they lose is a fairly strong indicator of madness.

@Icecreamdif If I did that, then yes. Gough is very chilling and in control, but it's possible. (I don't think Neil Cross would know the Toymaker if he jumped out of a tree, so it's down to whether you think Moffat snuck in a Toymaker reference.)

Well that Doctor has met many mad gods. And it could refer to the Post War Universe which was ruled by a Mad and Evil God King future Incarnation of the Doctor called the Emperor. But it could also refer the many other times such as before the anchoring or worlds of the Great Old Ones such as Fenric and of course Narlythotep (who is the Doctor and also incidentally a mad god). It probably does not refer to the Guardians such as the Black and White Guardians who amounts them only two are mad the Toymaker (the Crystal Guardian) and the Red (who is also the Doctor and incidentally a mad God.)

I think it could refer to the Time Lords

It’s a concept explored a lot in the show. He could be talking about Omega. Who knows? Select your favourite mad god king space daddy and jam it in if it makes you feel like it’s lore. The doctor has adventures you see and don’t as well. It sounds cool and Matt delivers it. The fact that we are speculating like this means the line did the heavy lifting it was supposed too :)

Omega is a good possibility as well

Rassilon in Zagreaus, the master every time he lures the doctor into the matrix to do battle, the valyard did something like that too. But that is also him.

A YouTuber who did a video where they put flashbacks and forwards during The Doctor's speech used Omega as the mad man god.

The Master of the Land of Fiction, Omega, the Celestial Toymaker, whoever ran that lab in Scherzo... Lots of people it could be.