Some of the most fun I've had playing dungeons is when things do not go smoothly and it forces you to do a non-standard run and think on the spot.
In the Stone Vigil the healer kept dying and I took over doing the heals as a black mage using acn's psysick x-class move. On the last boss, the healer and one dps died, so it was left to just the tank and me for half the battle. I had to spam psysick a lot on the tank because blm mind stat is low, then quickly use blizzard to get an umbral ice stack for mp recovery. We actually beat it with just two people and it felt extra rewarding. The same thing has happened to me on the chimera of cutter's cry.
I've also had a tank drop from the party in Toto-Rak at the very start of the dungeon and we did the whole thing as a 3-man party with me doing the tanking as an arcanist with topaz carbunkle. Our healer had to work extra hard, but we beat the whole dungeon without a tank somehow.
Share your own stories of glory against the odds.