jmc

An integer $a$ has an inverse $(\mathrm{mod}55)$ if and only if $\gcd (a,55)=1$. Similarly, an integer $a$ has an inverse $(\mathrm{mod}66)$ if and only if $\gcd (a,66)=1$.

Since we're looking for an integer that doesn't have an inverse modulo either $55$ or $66$, we want $a$ such that $\gcd (a,55)>1$ and $\gcd (a,66)>1$. Thus $a$ must be divisible by either $5$ or $11$, and $a$ must also be divisible by either $2$, $3$, or $11$. The smallest positive integer that satisfies both properties is $\boxed{10}$.