Does anyone make a chess clock that supports 3 or 4 players? 
Obviously I won't be using such a device to play chess, but instead to
play other games such as Scrabble.

Hi,
Surprising that Samtimer hasn't thought to make what you want.
What about a laptop solution?  I can do it, but surely someone else
can figure out how to link four clock faces or digital countdown
displays to a master timer so that it only operators one of the
displays at any time.

Try this http://www.123oy.com/g8/Game%20Timer.html
Search query: multiplayer countdown timer board games

This is a free comment.

That should be it, and cheaper than Samtimer's!
And this is a free  compliment to Crythias-ga!

Good find, crythias.  Ideally there would be a clock that shows all
the players' times at once, instead of just the current player's.  But
this will definitely do.  Thanks.