Let $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le M \text{ and } \rho(A) < 1\}$ where $M \ge 1$ is some fixed constant and $\|\cdot\|_2$ denotes the induced $2$-norm. Is it possible to give an upper bound $C$ in terms of $M$ such that $\|(I-A)^{-1}\|_2 \le C$ for all $A \in \mathcal E$?
As commented by Omnomnomnom, it is not possible to find an upper bound of $\mathcal E$. What if now we modify the family to be \begin{align*} \mathcal F = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le M \text{ and } \rho(A) \le r\} \end{align*} where $M \ge 1$, $0 < r < 1$ are fixed constants and $\|\cdot\|_2$ denotes the induced $2$-norm. In this way $\mathcal F$ would be a compact set. Since $\text{Inv}$ is continuous, $\|(I - \cdot)^{-1}\|_2$ must achieve maximum on $\mathcal F$. Can we characterize the upper bound in terms of $r, M$?
p.s. I asked a similar question here but formulated in a way wasn't intended and it was answered.
A partial answer.
Up to a change of orthonormal basis, we may assume that $A$ is triangular.
If $A$ is normal (then diagonal), then it is easy to see that the required upper bound is $\dfrac{1}{1-r}$.
Yet, we can do much better. Assume that $M$ is large (for example $M\approx 100$) and choose $A=rI_n+MJ$ where $J$ is the nilpotent Jordan block of dimension $n$.
Then $\rho(A)=r,||A||_2\approx M$; moreover $(I-A)^{-1}=\dfrac{1}{1-r}I+\cdots +\dfrac{M^{n-1}}{(1-r)^{n}}J^{n-1}$ and $||(I-A)^{-1}||_2\approx \dfrac{M^{n-1}}{(1-r)^{n}}$.
Thus, if $r=0.9,M=100$ and $n=7$, the upper bound is larger than $10^{19}$.