For a character $\chi$ of a finite group $G$,
$$|\chi(g)|\leq \dim(\chi)$$
Obviously the inequality is strict for $g=1$, and if $\dim(\chi)=1$ also for $g\neq 1$.
Is there a better bound for $\chi(g)$ if we assume 1) $g\neq 1$, and 2) $\dim(\chi)>1$?
No. For instance, taking $\chi$ to be a sum of $n$ copies of the trivial character for some $n>1$, then $\chi(g)=n$ for all $g$.
It also does not help if you assume additionally that $\chi$ is irreducible. For instance, let $G=D_8$ and let $\chi$ be the character of its usual two-dimensional representation as symmetries of a square. Then $\chi$ is an irreducible character of positive dimension. However, if $g\in G$ is the element corresponding to a rotation by $180^\circ$, then $g\neq 1$ and $\chi(g)=-2$.
(In fact, more generally, if $g\in Z(G)$ and $\chi$ is irreducible over $\mathbb{C}$, then $|\chi(g)|$ is always equal to $\dim \chi$. This is because the representation must send $g$ to a scalar multiple of the identity, by Schur's lemma.)