Personally happening to me, nothing that great. I got a flight canceled and had to stay the night in a comped hotel and flew out the next day. No big deal. However, the worst airport related story that I experienced sort of indirectly would be a few years ago.
My girlfriend at the time was flying back from a study abroad program in Italy. She was scheduled to arrive in Dallas Texas around 11 pm. We lived in 3 hours away in Oklahoma. It was cheaper to fly in to Dallas and have her parents pick her up. However, I get a call from her sister around 7 PM asking me if I could go get her. Something about it being late at night, and their mother wasnt so great at night time driving and Dallas can be a nightmare to navigate. So I agreed of course, and set out on the 3 hour drive. I was running a bit late, but was making good time on the highway until a large storm hit halfway through. It was dark and rain was pouring down hard. Traffic crawled down to 30 miles an hour. Luckily I had my pickup and wasnt too big of a problem, but there were a handful of cars pulled over on the side of the road till the storm cleared. I received a call from my girlfriend half way through, and she updated me on her status. She arrived in Atlanta, but there were storms throughout the south, and she was telling me her flight was delayed. I thought fuck, I was hauling ass through a storm to be on time for a delayed flight but no biggie. I slowed down and just talked to her for awhile as I made my way to Dallas. After we hung up, she would text me here and there because she was losing her voice, and as I got nearer to Dallas the flight was delayed by an hour... again, again, and again. By the time I got to the airport (11:30ish) her flight was delayed back to 2:30 am. When she called again, I told her at this point that they would probably just cancel the flight. I was tired of sitting at the airport for several hours so I said fuck it, and booked a hotel. The next time she called she informed me that they indeed canceled the flight. I told her no worries, I would be waiting for her the following morning to pick her up on the next flight out. However, there werent any flights to Dallas for the next three days. Ohhh fuck me. I called the airport and got a flight from ATL to OKC the next afternoon around 3 pm arrival time. I gave her a call and gave her the good news, and said I would pick her up back in OKC the next day. By the time all of this was done, it was around 3:30 am. I figured okay, shower, sleep, and drive all the way the fuck back to OK. Wrong, she called me not 10 minutes later saying she booked an earlier flight to arrive in OKC around 8 am. Well fuck. I took a shower and checked out of the hotel at 4 am, gassed up the truck and began my 3 hour drive back to OKC. 20 minutes away from the airport I get a call from her sister. She told me: Hey, you dont need to pick up Sara by the way, our dad is going to get her so you dont need to worry about it." FUCKING WHAT? That was my worst airport nightmare. There is more story but its irrelevant to the question.