What is the most sought after gramophone record?

I cannot say definitively as, with all antique and specialized items, it depends upon how much a collector is prepared to pay but I should think that one of the records made by Enrico Caruso in Milan in 1903 must be well at the top of the list. I collect old recordings, mostly 78rpm, and the last time I saw one in a catalogue it was priced at £600.

That's very true. It depends on the genre' of music and the genre' of the collectors. A big rule of thumb is a record is only worth what somebody is willing to pay for it.

In the R&B world of 78's - "Stormy Weather" by The 5 Sharps is probably the most sought after. Only 3 are known to exist and can demand a price of $20,000.00

But - there are high end country records and classical records. Depends on your knowledge and your taste.