  A classic result of Ritt describes polynomials invertible in radicals: they
are compositions of power polynomials, Chebyshev polynomials and polynomials of
degree at most 4. In this paper we prove that a polynomial invertible in
radicals and solutions of equations of degree at most k is a composition of
power polynomials, Chebyshev polynomials, polynomials of degree at most k and,
if k < 15, certain polynomials with exceptional monodromy groups. A description
of these exceptional polynomials is given. The proofs rely on classification of
monodromy groups of primitive polynomials obtained by M\"{u}ller based on
group-theoretical results of Feit and on previous work on primitive polynomials
with exceptional monodromy groups by many authors.
