The only "demigod" I know who would apply would be Dionysus in Greek mythology who had to prove his divinity but didn't need other gods to raise him up to godhood. Are there any others like him in any mythology?
Heracles is one of the most famous characters in Greek Mythology who transcended from being a demigod to a god. Throughout his life, Heracles faced constant trials, many of them concocted by Hera due to her hatred at Zeus' interactions with Heracles' mother (a mortal woman).
The story of Heracles' death begins with a centaur who attempted to kidnap Heracles' wife, Deianira, and was slain by her husband's wrath. In the centaur's dying breath, he gave Deianira the blood-stained tunic which he wore and told her that it was, essentially, a love potion for Heracles.
Years passed, and Deianira heard rumors that Heracles may have had a rival lover. Not knowing any better, Deianira gave her husband the tunic, thinking it would rekindle his love for her. Instead, the tunic (soaked in the centaur's blood which had been tainted by the Hydra's poison from Heracles' arrows) caused the skin to consume away on Heracles' body. He writhed in agony, and as a last resort to end the pain, he ordered a pyre be ignited to be burned upon.
As he died, he left his mortal body behind, and his immortal spirit drifted to Mount Olympus where he became fully divine. It may not have necessarily been his power that caused his transcendence, but it could very well have been the case. He wasn't raised by other gods, instead, he fought against them to earn his place among the Olympians because of of his strength and willpower.
See Also: The Death of Heracles