Yes. The Crab Opening. 1. a4 followed by 2. h4 (or vice-versa). I won some fun games with it :]
What openings bring out the rook(s) the fastest?

The rooks serve a very active purpose along the center files in this line.

Ruy Lopez gets the king rook going very quickly on the e file. I don't know of any opening that could do it faster without it being horrible. If a4 and h4 are intending to bring the rook out vertically, that seems like a pretty bad idea. The rooks are too clumsy to move around with so much stuff on the board, and if your opponent has all their pieces, the rook will just be attacked with limited movement while you fall behind on development. The rook will have more luck when some pieces are exchanged, or at least when you are developed and control the center.I won't deny that it could be a fun way to play, but there are obviously better ways to play by a lomg shot.

Probably not what you were expecting to, but I think that the Ruy Lopez activates the rook quite fast.
For example in the berlin variation
Or the Open variation

Off the top of my head, the only opening I can think of with an early rook activation is the Emory Tate variation of the Alekhine's defence, which is playable for white despite it's appearance.
This is just about the only time in chess where you can soundly make 2 rook moves before moving any of your other pieces.
Rooks are normally the last to the activated. Are there openings that get one or both rooks into the fight faster than usual?