jmc

First, we figure out how many nights it takes for every meerkat to stand guard with every other meerkat exactly once. There are $10$ possibilities for the first guard and $9$ possibilities for the second guard, making $10\cdot 9$ pairs; but this actually counts each pair twice (since it doesn't matter which guard is "first" and which is "second"). So, the number of nights in one complete period is $(10\cdot 9)/2$, which is $45$.

During that period, each meerkat has to stand guard for $9$ nights. So, each meerkat gets $\boxed{36}$ nights of sleep.

Alternative solution: Let's say we want to know how many nights of sleep a particular meerkat (let's call him Max) gets. This is equal to the number of pairs of meerkats that don't include Max. To form such a pair, we can choose the first (non-Max) meerkat in $9$ ways and the second in $8$ ways, but again, this overcounts by a factor of $2$. The number of pairs without Max is thus $(9\cdot 8)/2$, which is $\boxed{36}$.