Why did Prince Philip renounce his Danish and Greek royal titles to marry the...

[Regina]

...future Queen?
He didn't have to but he did so to improve his profile...?

[lili]

Because his Danish and Greek royal titles didn't actually mean much at the time. He wasn't in line to inherit either title -- he was distant from them -- the Greek throne had always been unstable and uncertain, and he had found a home in the UK, especially in the British Navy. He'd been largely educated in the UK, too, and had supportive relatives there, because like almost all members of royal families in Europe at the time, he was heavily British in terms of descent and relationships.

Marrying the future Queen Regnant of the UK was absolutely the best he was going to do for himself. He had no chance at the thrones of either Greece or Denmark, since he was such a minor prince.