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1.
Going ridiculously fast, preferably down hill.
So i was on my sled, right, and i was barrel-assing down the hill...

or

Woah.. did you see that Viper go barrel-assing past us?
by Goldphishe August 24, 2004
 
2.
Driving at high speed through an area not intended for vehicles.
How did you break your arm?

We were barrelassing through the field and rolled the go-kart.
by Iridescent Shadow December 30, 2008
 
3.
The act of driving very fast on a road or highway or street. Can also be applied to the act of pursuing a personal or professional goal. Originated in the 1960's, originally used in Brooklyn NY, describing a return trip from the Catskills in Upstate NY.
"I had to be back to light the boiler at the plant, so here I am barrel assing down the highway right past a cop." "He is determined, I mean he's barrel assing toward his goals."
by Charlie 1970 to Joey The Kid December 18, 2011