Most LAMEs I've known have been pilots and owned a plane and have been happy to fly to help recover broken planes.
One Avionics LAME regularly used his plane to fly around the country doing jobs for people.
An advantage of being a LAME is the ability to accumulate a collection of free/cheap equipment to fix up planes they bought cheaply.
Whenever a fuel system drain and calibration is done on your plane new fuel is always used to refill the tanks as a "safety" measure. The old drained fuel goes into the LAMEs own plane. Better than wasting it I guess :-(