This tower of fields is being ridiculous

Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2).

Now obviously $[\mathbf{C}:\mathbf{R}]=2$. But consider the fact that the algebraic closure of $\mathbf{R}(t)$ has cardinality $2^{\aleph_0}$---this implies that $\overline{\mathbf{R}(t)}\cong\mathbf{C}$, so in particular we can embed $\mathbf{R}(t)$ into $\mathbf{C}$.

If we embed $\mathbf{R}$ into $\mathbf{R}(t)$ in the natural way, we get $$\mathbf{R}\subset\mathbf{R}(t)\subset\mathbf{C}.$$

So our good fact at the beginning would have us believe

$$2=[\mathbf{C}:\mathbf{R}(t)][\mathbf{R}(t):\mathbf{R}].$$

What is the meaning of this? Either these two degrees really are both finite or (more likely) I've made a huge mistake. Perhaps it would all be clear if I were more precise about "embedding" $\mathbf{R}(t)$ in $\mathbf{C}$.

Your mistake is that $[\mathbb{C} : \mathbb{R}] \neq 2$!

To define the degree of a field extension is not enough to know the two fields involved (except in special cases): you actually have to know what the field extension is. In this case, the field extension $\mathbb{R} \to \mathbb{C}$ you constructed is not the field extension that comes from the inclusion $\mathbb{R} \subseteq \mathbb{C}$, and therefore it can, and does, have different degree.

A simpler (and more dramatic!) example of this phenomenon is the field extension $F(x) / F(x)$ given by the embedding $F(x) \to F(x)$ that sends $x \to x^2$. In this case, we have

$$ [F(x) : F(x)] = 2 $$

The supposition "$K \subseteq F \subseteq L$ as fields" usually implies more than it says: it also implies that the letter $F$ will sometimes be used not for a field, but for the field extension defined by the inclusion $K \to F$. Occasionaly we might disambiguate by writing $F/K$ rather than $F$. Similarly, $L$ will sometimes mean a field, and it will sometimes mean $L/F$ and it will sometimes mean $L/K$.

These sort of technicalities is the price we pay for the greater flexibility of allowing extensions to be any injective map $F \to E$, rather than requiring field extensions to come from actual subset relationships $|F| \subseteq |E|$ among sets. (In that last expression, $|F|$ means the underlying set, and $\subseteq$ has its usual set-theoretic meaning)

P.S. if you're ever in the situation where you consider the field extension $F(x) \to F(x)$ above, do yourself a favor and rename the indeterminate variable in one of the two copies of $F(x)$ rather than blithely forge ahead as I did above for dramatic effect. Similarly, it's probably wise to add a decoration to $\mathbb{R}$ to indicate when you are using it in a way inconsistent with its canonical inclusion into $\mathbb{C}$. (or decorate $\mathbb{C}$)