My story is not so much about what is forgotten at check in but rather at checkout.
Rostered , early 1990's, to return a 737-200 back to the owner, a leasing company, from Hong Kong to Oklahoma City via the long way round. After 4 great days, 3 overnights we arrived at OKC Will Rodgers, having come from last stop Sept Iles, Quebec, 2 Aussie pilots and one Brit riding engineer, we were directed by ATC to the ILFC aprons and hangars for parking. We were then required to pack up all company documents on the aircraft for shipment back to HKG before leaving, including Jepp charts, pax briefing cards, QRH, checklists, tech log, etc etc. This took awhile and when completed handed these boxes to ILFC for shipment back to the company. We then piled into a Cadillac and driven off the airport through a back gate and to a nearby hotel.
Next morning we got a call from ILFC to say the US immigration were looking for us and we needed to appear pronto at their offices in downtown OKC. We had completely forgotten to clear Customs and immigration and now there was an APB out. Duly arriving at the very crowded immigration office we were told to take a number and wait to be called, several attempts were made to tell the lady at the enquiry desk about our case but she insisted we take a number and wait. 2 hours later we were called and it was a comedy scene. They were terrific about it and all had a good laugh.
I wouldn't want to do it in this day and age.