It's a standard lemma that the gcd of the binomial coefficients $(k,n-k)$ (for $0\lt k\lt n$) is $p$ when $n=p^r$ for some $r>0$ (with $p$ prime) and $1$ in all other cases.  It follows that for any $f(t)\in R[t]$ with properties as described, there is a finite splitting $R=\prod_{i=1}^mR_i$, where for each $i$ either

(a) $f(t)$ maps to $t$ in $R_i[t]$; or

(b) Some prime $p$ is zero in $R_i$, and $f(t)$ maps to $t^{p^{r_i}}$ in $R_i[t]$ for some $r_i>0$.

Thus, the general case is not very different from the ones you mentioned already.  These ideas crop up in the study of formal group laws and the dual Steenrod algebra, so they are fairly well known among algebraic topologists.

UPDATE: as darij says, for the centre of $R$-Alg we should additionally insist that $f(at)=a f(t)$ for all $a\in R$, or equivalently that $a^{p^{r_i}}=a$ for all $a\in R_i$.  In the case where $r_i=1$ and $p=2$ this means that $R_i$ is a Boolean algebra, so by the Stone representation theorem it is the ring of continuous functions from $X_i$ to $\mathbb{Z}/p$ for some Stone space $X_i$.  I think this also holds for $p>2$.  If $r_i>1$ then we can still pick a field $F$ of order $p^{r_i}$, whose Galois group $G$ over $\mathbb{Z}/p$ will be cyclic of order $r_i$.   If $X_i$ is a Stone space with an action of $G$, then the ring of continuous $G$-equivariant maps from $X_i$ to $F$ will have the properties required for $R_i$.   I think that every possible $R_i$ arises in this way.