Can I Successfully Be A Photographer Without Getting A Degree?

It depends to some extent on the level of success you are seeking.  If you want to run a successful high street photographers then it is likely that you could do this without a degree, although some kind of qualification would give you some credibility.  So if a degree is not for you, then even an evening class or college diploma would be useful qualification.

If you want to be an international photographer on the scale of David Bailey or the likes, then a degree would certainly be very useful for you.  This is because it can help you to prove to employers (and realistically you will have to be employed at least initially) that you are serious, that you have already been taught photography to a high standard and that you are capable.
Sadly these days, if you do not have a degree there will be about 10 other applicants who do, so it is always best to prove your commitment by studying hard !