Nice discussion! So we can say that for any unital subring $D$ of $\mathbb{R}$, any homomorphism from $D[[X]]$ into $\mathbb{R}$ has the form $\varphi(a_0+a_1X+\cdots)=\zeta(a_0)$, where $\zeta\colon D\to\mathbb{R}$ is a homomorphism. Further, the only homomorphisms from $\mathbb{R}[[X]]$ into $\mathbb{R}$ are the zero map and the map sending every power series to its constant term.

Very nice! Thanks.

ah yes, of course. Thank you!

Oops - thanks for this and sorry for the delay.

I see that a ring with a finite maximal subring must be finite, and so this answers the question.