If $K$ is a number field, is it always possible to find a finite extension $L/K$ such that $L$ is the composite of fields $L_1,\ldots, L_n$ with each $L_i$ ramified at at most one finite place? Equivalently, is the composite of all extensions ramified at $\leq 1$ place all of $\overline{\mathbb{Q}}$?

The first question has an affirmative answer when $K$ is abelian, but for the general case, the equivalent second question sounds too strong to be true. Any ideas?

If $K$ is a number field, is it always possible to find a finite extension $L/K$ such that $L$ is the composite of fields $L_1,\ldots, L_n$, with the property that at most one prime ramifies in $L_i/\mathbb{Q}$? Equivalently, is the composite of all extensions ramified at $\leq 1$ place all of $\overline{\mathbb{Q}}$?

The first question has an affirmative answer when $K$ is abelian, but for the general case, the equivalent second question sounds too strong to be true. Any ideas?