Suppose $L/K$ is a field extension of degree $p^n$ for some prime $p$ (if necessary, assume the characteristic of $K$ is not $p$).
Then, is it always possible to find a sequence of extensions $K = K_0 \subset K_1 \subset K_2 \dots \subset K_n = L$ such that $[K_r:K_{r-1}] = p$?
Using Galois theory, this problem translates into the following:
Suppose $G$ is a finite group with a subgroup $H$ such that $[G:H] = p^n$. Is it always possible to find a subgroup $G \supset H' \supset H$ so that $[H':H] = p$?
No, this is not always possible. For instance, consider $G=A_4$. Then $G$ has a subgroup of index $2^2$ (any subgroup generated by a $3$-cycle) but has no subgroup of index $2$.