Probably any time a "boss" song is first passed on its highest available difficulty. Special mention goes to Drop Out (the first 9), the entire MAX series (extra stage, no re-charging life bar), Pluto Relinquish (sudden death mode), and Valkyrie Dimension. (also Sudden Death mode).
The two marathon world record attempts of note under Guiness World Records rules (must start at beginner, progressing up through the difficulties at various intervals, up to 15 minute break per hour, must pass every song, no bar, no more than 30 seconds between songs):
Then in 2011, Travis Houk went Up to Eleven going not only past both the official 14 hour record and the infamous 17 hour record, but going for forty hours (the official time was 40:02:12 exactly) on DDR X2. Ceremoniously, he saved MAX 300 for the last song, which he did indeed clear. Then, almost intentionally, he went straight to Valkyrie Dimension on Challenge. Which he promptly failed.
How does Mario (or Luigi) defeat Waluigi, Wario, and Bowser and recover the Music Keys? By facing against them in a dance off! Bowser break dancing is both hilarious and awesome at the same time.
The Background Video of LOVE IS THE POWER Re:born, showing exactly what the DDR Community has accomplished in DDR X3 vs 2nd Mix.
Said song, in Encore Extra Stage form, requires you to pull an Awesome Moment or else you get failed out. Specifically, you need to get a Perfect Full Combo.