It only took me 14 years (I started playing when I was 5), but I finally did it.
I thought this would take some time, so to prevent the dreaded Nintendo Thumb from kicking in, I took the shortest route I could: Take the Warp Zone in level 1-2 to level 4, then taking the Warp Zone in level 4-2 to level 8.
I grabbed a Fire Flower early on and blazed through world 8 (fucking Hammer Bros.) before I died for the first time in the castle. There were no Super Mushrooms in the castle, so I was stuck as regular Mario for the last stretch.
After clearing that goforsaken pipe puzzle (figure out which warp pipe takes you where) and going through that bizarre, short swimming segment, I was finally face to face with King Bowser himself, standing on a bridge over a pit of lava. There, at the other end of the bridge, was an axe that would cut the bridge and send Bowser plummeting to his death
There was no easy way to get to that axe; Bowser was jumping around, spitting fire and throwing hammers at me with reckless abandon. I only had one life left by this point, so I had to make this count. I charged for the axe, but Bowser threw a myriad of hammers and nearly hit me, so I backed off towards the edge of the bridge. He jumped again and I made another mad dash, but Bowser wasn't a fool, no, no. Just before I ran under him, he let loose a stream of fire that would have ended me, so I, again, retreated to my end of the battlefield.
"Surely, there must be some way I can get to that axe," I thought to myself. But alas, there was only one way: To make a mad dash for it and hope I don't get killed.
With no other choice, I raced for the weapon and, through some stroke of luck, timed it just right, so I jumped over Bowser and landed on the axe, cutting the bridge and sending Bowser plummeting to his fiery death.
The quest over, Mario finally rescued the Princess and got laid, I guess. I dunno.
Tl;dr: I beat Super Mario Bros.