jmc

An integer that is congruent to $15(\mathrm{mod}22)$ is of the form $22n+15$.

Therefore, we make the equation $22n+15<10000$, and find the largest possible $n$. $$\begin{array}{rl}22n+15& <10000\\ 22n& <9985\\ n& <\frac{9985}{22}\approx 453.85\end{array}$$The largest possible integer $n$ is $453$. We plug it in for $n$ to get $22\cdot 453+15=\boxed{9981}$.