How to find recurrences where Master formula cannot be applied
For the recurrence, 
$$
T(n) = T(\sqrt{n}) + 1
$$
Let $n = 2^{2^{k}}$, therefore we can write the recurrence as:

$$
T(2^{2^{k}}) = T(2^{2^{k-1}}) + 1 \\
T(2^{2^{k-1}}) = T(2^{2^{k-2}}) + 1 
\\\ldots\\\\\ldots\\
T(2^{2^{k-k}}) = 1 
$$

, i.e. $ k * O(1) $ work or linear in $k$. We can express $k$ in terms of $n$:
$$
\log{n} = 2^{k} \\
\log{\log{n}} = k
$$

Hence, the recurrence solves $T(n) = O(\log{\log{n}}) $