Probably not as many as you think. Since I didn't have to bother with fighting monsters, and the main goal was just to go down and not get trapped while doing it, I could easily go down 5-10 levels at a time in a few minutes by starting in or near rooms with the stairs or hitting pit traps early. Finding ways out of situations like floor 182 took a while, and having to explore mazes, but overall just run run run dive run run dive dive, etc. If you look at the leaderboard, I used 75,000 turns and collected 300,000 gold. Second place used 48,000 turns and only collected 80,000 gold. I also killed 100 less monsters.
I'll tell you what took forever though, starving to death. I must have fainted 100 times before he finally died, it was ridiculous. I should have taken the screenshot of 10+ monsters lined up on each side of me on the minimap.