Is there a number that is exactly 3 more than its cube?

Yes, there are three of them.  Two of them are complex.

They are the solutions to (x-x^3=3).

The exact value for the real solution is:
(sqrt(239)/(2*3^(3/2))-3/2)^(1/3)+1/(3*(sqrt(239)/(2*3^(3/2))-3/2)^(1/3))

Approximately, it is x=-1.67169988...

If you are expecting integers (which was not specified), then there is none.