01 811 8055
This used to be the BBC number for call ins, particularly the kids TV show 'Swap Shop', but also for so much else during the 1970s and 1980s.
This number was retired in 1990 when the London ran out of phone numbers and switched to two different prefixes, 071 and 081. The former was advertised on TV as 'Inner London' and the latter as 'Greater London'. This bit of marketing kept everyone happy.
There was still a problem with numbers and the need to go for eleven digits. Hence, in 1995, the codes for London changed again, to 0171 and 0181. This was PHONEDAY.
But still, more numbers were needed, plus the tech behind the scenes was ever-evolving. Hence, in 2000, the numbers changed again for London, for everything to start with 020, so 0171 became 0207 and 0181 became 0208.
But then everyone got mobile phones and we no longer heard about how the economy was growing so quickly that we had this apparent incessant need for even more phone numbers. Furthermore, mobile phones had contacts built into them, so there was no need to remember phone numbers, which was just as well as eleven digits were not so easy to memorise, particularly when the prefixes had changed around so much.
Hence, my personal choice of fictional number. Apart from anything else, it enables me to see how well forms are validated, plus 01 811 8055 is only going to ever be recognised as a 'famous' number by Brits over a certain age.