I spent a while flying a PA18 with no transponder. I was never denied a clearance anywhere in that time.
There was only one thing which caused me problems, and it was invariably Farnborough Radar who were responsible. Farnborough, for those who don't know, are a fantastic LARS station, who very often radar-identify you even just on a FIS, and frequently provide traffic information even when it's not required. I soon discovered, though, that if I asked for a FIS without a transponder, they would still try to keep tabs on me just as if I was squawking. Which meant they were asking me for frequent position reports - often at places I'd never heard of, which required some heads-in time to locate the place on my chart. I soon learnt not to bother with FIS, but there are times when it's not possible to avoid it, such as when you need a MATZ penetration or a DAAIS.
FFF
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