Doing my stats thang again. According to our database there are only 2 Pokemon that learn a move of every type: Mew and Drampa.
Smeargle didn't show up there because technically Sketch is the only move it actually "learns". But of course, Sketch can copy moves of all types, and those moves are there permanently so I think it counts. Personally I wouldn't include Arceus or Silvally for Judgment/Multi-Attack because then you should include Hidden Power which practically every Pokemon can learn.
If you're interested, there are 24 Pokemon that learn moves of 17 types:
Nidoqueen, Clefairy, Clefable, Persian, Mankey, Primeape, Kangaskhan, Mr. Mime,
Togetic, Teddiursa, Ursaring, Lugia,
Aggron, Latias, Rayquaza, Jirachi,
Togekiss, Gallade, Uxie, Mesprit, Azelf, Giratina, Arceus,
Most of them are missing a Fairy type move.