Can a Boeing 747 or A380 climb with only one engine?
I doubt that either aircraft can climb on a single engine (though momentum and airspeed could help, as @RonBeyer already commented). The closest an aircraft came to this condition was the British Airways Flight 9 when it ingested volcanic Ash. 

During this incident, the aircraft flew on one engine for around 90 seconds. In this case, a report from the flying magazine says:


  On one engine they could maintain height at 13,000 ft; over the next five minutes, the next three engines relit, and they begun a climb.


This seems to indicate that the aircraft was not able to climb with one engine. However, during this incident, the aircraft was not climbing, but was descending at 1800m per minute.

Note: There seems to be some confusion about the altitude. The captain's website gives the altitude as 12,000 ft.