The Boeing 777 APU can be started and provide electrical power up to the maximum certified altitude of the 777.
Bleed Air is available up to FL220.
The air turbine starter (ATS) will automatically engage when there is sufficient bleed air available - otherwise the electric starter will be commanded on. The electric starter has a dedicated battery. 2 starters = redundancy.
Whether the APU runs on takeoff is at the Captain's discretion or subject to the airlines' SOPs.
EDIT: Mr. Mutt, I didn't know of the FLO engineer's advisory. Bulletin I guess ? I stand corrected here.
Something you didn't ask but I offer:
the 777 APU will autostart when airborne if all 4 generators (main and backups) fail.