What is the time complexity of this atrocious algorithm?
We can write a recurrence relation for this procedure as follows. Let $T(n)$ be the worst-case time for running sort on a list of length $n$. When calling sort on a list of length $n$, we can have up to $n$ recursive calls which rotate the list to the left until its minimum reaches the first entry. Each such call involves an invocation of minimum, and so takes $T(n-1) + O(n)$. The final one descends to sorting a list of length $n-1$. In total, we get the recurrence
$$
T(n) = n T(n-1) + O(n^2).
$$
Opening this recurrence, we get
$$
T(n) = O(n^2 + n(n-1)^2 + n(n-1)(n-2)^2 + \cdots + n!) = O(n!).
$$
(We get $O(n!)$ since the terms decrease very quickly from $n!$ to $n^2$, faster than a geometric series.)

Of course, without giving a matching lower bound, we don't know whether this analysis is tight.