Perhaps this is a difference between British and American English because the examples that Jimmy gives would be awkward in the U.S. Of course everybody knows what you mean but in the U.S. nobody would say, for example: "I will get to look older when my hair is white." OR "I am now getting to be an old man." From my perspective, I agree with the first answer. "Got to be" is more or less a synonym for "have to be". For example: "I am a doctor so I've got to be 100% certain that I give patients the correct medicine." = ""I am a doctor so I have to be 100% certain that I give patients the correct medicine." But you can't say: "I am a doctor so I've got to become 100% certain that I give the patients the correct medicine." I don't know if that last sentence works in British English but in the U.S. you would never hear it.

"got to" is a very colloquial way of saying "have to" (müssen), usually pronounced "gotta" (and sometimes even spelled that way in informal writing like text messages etc.). It's very common in speech but not really used in writing (and should be avoided in any formal text). For example: "I got to be somewhere at 5pm." == "I have to be somewhere at 5pm" "I got to go!" == "I have to go!" "You got to see this! == "You have to see this!"

You are right, though the synonym for 'become' here is 'get'. Your example uses 'to be' as the verb - after years of hard work, I got to be rich. I am now getting to be an old man. You can use other verbs too: I will get to look older when my hair is white. 'Get' works by itself, without another verb: I am getting old now, I got very hungry today. The only real difference between get and become is that if you use it with a noun (become an old man) you must add 'to be'.