What is the smallest class of reductions under which there is a $\mathsf{P}$-complete problem?
It is easy to show that the Circuit Value Problem is complete for $\mathsf{P}$ 
under $\mathsf{AC^0}$ reductions (see  András's comment below).

For an easier example consider 
$$A = \{\langle M,x,y\rangle \mid \text{$M$ accepts $x$ in $|y|$ steps} \}$$

If a class of reductions $C$ contains constant functions, pairing of strings, and functions where the size of their output can bound any polynomial;
then $A$ is complete for $\mathsf{P}$ w.r.t. $C$.