I will forever wish that I could say Super Mario Land on the Gameboy, but, after a nearly 14 hour drive from High Prairie to Kamloops I could never finish the final level even though I could get there with 1 continue worth of points.
My first game finish was Fist of the North Star on Gameboy. I still own the cartridge but the gameboy is in my dads master bathroom with a religiously unremovable Tetris.
I watched my older brother finish Zelda: Ocarina of Time.
I tried to do it myself at the time at would get lost or stuck constantly and quit.
I may have finished Star Fox on the 64 by myself, then as a teen I finished Twilight Princess.
My favorite games with no though... Dr. Mario, Balloon Fight, and Yoshi (the stacking game)
At least I don't know if they end.
The original Dizzy game (on the ZX Spectrum no less!), which apparently had the full title of "Dizzy – The Ultimate Cartoon Adventure" is the first vague memory but I think my dad helped a lot. The first game I know for certain I completed without any help was the original Alien Breed game. I loved that series!