Does anyone make a 3 circuit switch?
Yes, Pass &amp; Seymour Despard should git-r-dun.   

You have a much bigger problem though. That's an awful lot of wires. 

Assuming you have 3 supply cables and 3 onward cables, all /2, that means 12 conductors.   +2 for the yoke, +1 for all grounds, and +1 for all cable clamps, totals 16 conductor units. 

With #14 wire that will be 32 cubic inches.   With #12 wire that will be 36 cubic inches. 

A single box is about 10-12ci, so that ain't gonna happen.   A 4x4 2-gang box is 22 ci, and normal drywall 1-gang mud rings don't add enough cubes to save you.  They do make 10.2 ci. mud rings, but those won't work with drywall.  

You are better off going to a 4-11/16" square box which 42 cubic inches depending on depth.  That with a 1-gang mud ring will suffice.  

Now if you use all switch loops, you have 9 conductors, so with yoke+ground+clamps that is  13 wire counts. with the yoke+ground+clamps.  That's 26 c.i. For #14 wire and 29.25 c.i. with #12.  That might barely scootch under with a 4x4 box and a 1-gang mud ring if #14.