We have $a\star b = a^b+ab=15$.
Since $a$ and $b$ are positive integers greater than or equal to 2,
we can start by trying different values of $a$ and $b$.
If $a=2$, then $2^b+2b=15$.
Trying different values of $b$, we see that $b=3$ is a solution, since $2^3+2(3)=8+6=14$.
Therefore, $a+b=2+3=\boxed{5}$.
The answer is: 5
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Let $a\star b = a^b+ab$. If $a$ and $b$ are positive integers greater than or equal to 2 and $a\star b =15$, find $a+b$.

### Answer:
