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1.
To grill and relax at the same time
I'm going to put the burgers on, have a cocktail, and grillax
by Nexter Level March 24, 2009
 
2.
Grillax is a updated version, and a higher state of being, than "chillax". It happens when you're grilling or there's a BBQ involved in the fun you're having. You're already chillaxing (chilling out and relaxing) in the afternoon with your friends, but someone decides that there should be some BBQ involved. You're now one step above chillaxing, you're now grillaxing!
What do we do on Labor Day and July 4? We grillax! Put the burgers on the BBQ, grab some beers, and let's grillax! Grillaxing is what we do when we get together!