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1.
Fast slang, or the Ebonic translation of the request, "Let me get..." Most often used when ordering food, drink, or other such accommodations. Also used boldly when courting the opposite sex.
Ayy maaan, lemme git a rodeo cheeseburger wif a small fry an' a soda.

or

Ayy shawty, you fine as a muh, lemme git dem digits.
by Alan Heat May 11, 2008
 
2.
To ask to receive, awesome, an action, event, thing so incredible that you must have it for your own even if it is not obtainable physically.
Alex: Dude, Gabe is number one!
Keith: Dude, LEMMEGIT!!!

Myles: That guitar is so dirty!
Aaron: Yeah it is! Lemmegit!!!

by DirtyJap December 04, 2007