jmc

For $f$ to have an inverse function, it must not take any repeated value -- that is, we must not have $f(x_1)=f(x_2)$ for distinct $x_1$ and $x_2$ in its domain.

The graph of $y=(x+2{)}^2-5$ is a parabola with vertex at $(-2,-5)$:

The axis of symmetry is the line $x=-2$, so for every $x$ less than $-2$, there is a corresponding $x$ greater than $-2$ where $f$ takes the same value. If we restrict the domain of $f$ to $[-2,\infty )$, then $f$ has no repeated values, as $f$ is increasing throughout its domain. But if we restrict the domain to $[c,\infty )$ where $c<-2$, then $f$ has repeated values. So, the smallest $c$ which will work is $c=\boxed{-2}$.