Let $U_1, U_2,\ldots$ be a sequence of independent uniform (0, 1) random variables, and for $1 \leq k \leq n$ let $\xi_p(n, k)$ denote the $p$th quantile, $0 < p < 1$, corresponding to the block $U_{n -k + 1},\ldots,U_n$. In this paper we investigate the a.s. limiting behavior of $\xi_p(n, a_n)$ when $a_n$ is an integer sequence, $1 \leq a_n \leq n$, and $\lim_{n \rightarrow \infty}a_n/\log n = \beta \in \lbrack 0, \infty\rbrack$. In addition, we investigate the a.s. limiting behavior of $\max_{a_n \leq k \leq n}\xi_p(n, k)$ and other maxima involving the $\xi_p(n, k)$'s.