Most of the game's drama was a direct result of your sidekick. You played Lester Knight Chaykin, a scientist who finds himself transported to an alien world after an experiment goes awry. Taken prisoner by the locals, he's trapped in a cage with a large, hulking alien. This alien (who remains nameless throughout the game but who is reportedly named "Buddy") became instrumental in helping you escape.
The alien led you through the compound, helped you fight the guards, and from time to time got separated from you. He would always reappear, though, often just in time to save you from overwhelming odds.
The most dramatic events took place at the finale, when Lester is crippled and the alien returned just in time to whisk him away to safety. The strange, inconclusive ending seemed to signal a sequel, but it never came. Lester and the alien simply flew away on a huge winged beast, but Lester still had no way to return home.
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There was a sequel to Out of this World, a game called Heart of the Alien that was only released for the Sega CD. This game told the same events but through the eyes of the alien. It also featured a different, and more conclusive, ending. Unfortunately, it was never released for any other consoles.
A sidekick who drives in style