For me, the level I've spent most time on was Battletoads' Clinger-Winger.
Not that the level in itself is so impossibly difficult, but shortly after the NES Classic (NES-Mini) came to market, it was a great opportunity to bring a NES with me to work (I worked on a ship at that point, and the rotation was 6 weeks onboard / 6 weeks home). Loved the opportunity the NES classic brought, with compact size, lightweight, USB power and connection to modern TV's. Only problem was, that the screen in my cabin was some dated and cheap PC monitor, and of course, that Battletoads was emulated, so it was nearly impossible to time the corners correctly to get through that level. I played for hours and hours for days and days and had to learn to time the corners the correct way based on the lag. At the end I'm sure I could've done the majority of the level blindfolded considering how many hours I spent on it total. Horrible experience, but also rewarding.
Got back home, put a proper cart in my NES on a proper CRT and all of the suddenly it felt like a completely different game.
So, yeah, since then, I've joined the choir of people preaching that emulating is bad