Hilbert's hotel (HH) is only a metaphor, and when pushed too far it can lead to confusions. I think this is one of those situations: the key point is "we can't obviously compose infinitely many functions," which is pretty clear, but it's obscured by the additional language.

The point of HH is to illustrate how an infinite set (the set of rooms) can have lots of maps from itself to itself ("person in room $n$ goes to room $f(n)$") which are *injective* ("no two different rooms send their occupants to the same room") but *not surjective* ("some rooms wind up empty"). Note that already we can see an added complexity in the metaphor: the statement

There is a set $X$ and a map $f:X\rightarrow X$ which is an injection but not a surjection

has only one type of "individual," namely the elements of $X$, but HH has *two* types of "individual," namely the *rooms* and the *people*.

Now let's look at the next level of HH: getting an injection which is *far from* a surjection. Throwing aside the metaphor at this point, all that's happening is *composition*. Suppose $f:X\rightarrow X$ is an injection but not a surjection. Pick $x\in X\setminus ran(f)$. Then it's a good exercise to check that $x\not\in ran(f\circ f)$, $f(x)\not\in ran(f\circ f)$, and $x\not=f(x)$.

What does this mean? Well, when we composed $f$ with itself we got a *new* "missed element," so that while $ran(f)$ need only miss one element of $X$ we know that $ran(f\circ f)$ is missing two elements of $X$. Similarly, by composing $n$ times we get a self-injection of $X$ whose range misses at least $n$ elements of $X$.

At this point it should be clear why we can't proceed this way to miss an infinite set: **how do we define "infinite-fold" compositions?** This is what the question "where should the guest in room $1$ go?" is ultimately getting at.

It's worth pointing out that there *are* situations where infinite composition makes sense. Certainly if $f:X\rightarrow X$ is such that for each $x\in X$ the sequence $$x,f(x),f(f(x)), f(f(f(x))),...$$ is eventually constant with eventual value $l_x$, then it makes some amount of sense to define the "infinite composition" as $$f^\infty:X\rightarrow X: x\mapsto l_x.$$ And if $X$ has some additional structure we might be able to be even more broad: for example, when $X=\mathbb{R}$ we can use the *metric structure* (really, the *topology*) and make sense of $f^\infty$ under the weaker assumption that the sequence $$x,f(x),f(f(x)), f(f(f(x))), ...$$ converges (in the usual calculus-y sense) for each $x\in \mathbb{R}$. For example, the function $f(x)={x\over 2}$ would yield $f^\infty(x)=0$ under this interpretation *(even though only one of the "iterating $f$" sequences is eventually constant - namely, the $x=0$ one)*.

But **this is not something we can do in all circumstances**, and you should regard the idea of infinite composition with serious suspicion at best. (Although again, there are situations where it's a perfectly nice and useful idea!)