Let $A(x),B(x),C(x)$ be distinct univariate polynomials with rational coefficients of resp distinct degrees $a,b,c$. Also $a,b,c$ are odd and larger than 1.
Let ** denote composition.
We want the polynomials to be commutative with each other ;
So the compositions are
$$A(B(x)) = B(A(x))$$ $$A(C(x)) = C(A(x))$$ $$B(C(x)) = C(B(x))$$
Also, there exists NO univariate polynomial with rational coefficients $D(x)$ of even degree >3 such that D commutes with one of A,B or C.
( Ofcourse $D(x)$ not equal to A,B or C )
What are examples of such polynomials ??
Notice sin( degree arcsin(x) ) does not work and neither does x^degree ; since they do have a D(x).
So the Chebyshev polynomials are NOT what we want.
By a the theorem of Ritt-Julia (see http://nyjm.albany.edu/j/2007/13-5p.pdf) for commuting (non-linear) polynomials P,Q either they are similar to monomials or they are similar to Chebychev polynomials or:
$P^{\circ m} = Q^{\circ n}$
So lets choose A some nonlinear, non-monomial and non-Chebychev polynomial and set $B=A^{\circ 2}$ and $C=A^{\circ 3}$, so they obviously commute. If there is an even degree D that commutes with A there need to exist m and n such that $A^{\circ m} = D^{\circ n}$, but then $\deg(A^{\circ m})=\deg(A)^m$ which means that it is uneven while $\deg(D^{\circ n}) = \deg(D)^n$ is even.
So there can not exist such a polynomial D (similarly for $A^{\circ 2}$ and $A^{\circ 3}$ which again are not monomials and not Chebychev polynomials).