Why is one "$\infty$" number enough for complex numbers?

There is such a thing as a "one-point compactification" of $\mathbb{R}$; you end up with something that is "like" a circle (where "like" can be made very precise) with a point corresponding to $\infty$.

This is in exactly analogy to viewing the complex numbers with $\infty$ added as a sphere: you do it by stereographic projection. Take a unit circle and put it on the plane so that its center is at $(0,1)$. It is tangent to the $x$-axis at $(0,0)$. To each real number $r$ there corresponds one and only one point in this circle, obtained by taking the line through $(r,0)$ and $(0,1)$, and identifying $(r,0)$ with the second point of intersection of the line with the circle (the first point being $(0,1)$). Every point on the circle except for $(0,1)$ itself corresponds to a real number, so you can think of the point $(0,1)$ as corresponding to "the point at infinity". In fact, this is a natural way of constructing the real projective line.

The counterpart of this is that you can "add infinities" to the complex plane the same way that we "add infinities" to the real line, one for every direction. In the real line, we have the "positive direction" and the "negative direction", which lead to a $+\infty$ and a $-\infty$, respectively. In the complex plane, you would want to add an "infinity" in every direction, so you would have to add an $+\infty_m$ for every real number $m$ that corresponds to the direction from $0$ to $1+mi$, and a $-\infty_m$ corresponding to the direction from $0$ to $-1-mi$ (in the "opposite direction" of $+\infty_m$); plus a $+\infty_v$ corresponding to the direction from $0$ to $i$, and yet another $-\infty_v$ for the direction from $0$ to $-i$. Doing this essentially gives you a closed disc.

Yet a third way is to consider adding a point for every slope, adding an $\infty_m$ corresponding to a line of slope $m$, plus an $\infty_v$ for the vertical lines on the complex plane. In this case, what you get is essentially the real projective plane.

You can do any of the three, but each gives a different kind of structure. Just like the extended reals (with $+\infty$ and $-\infty$) is (usually) the "right" setting to do a lot of calculus, rather than trying to do it in the projective real line, so the Riemann sphere (obtained by adding a single $\infty$ to $\mathbb{C}$) is (usually) the "right" setting to do complex analysis, rather than trying to do it in the closed disc or the projective plane. That is, you "can" any of them, but the "one $\infty$" completion of the complex numbers is more useful for analysis (and in other settings) than the "one $\infty$ per direction" completion or the "one $\infty$ per slope" completion. If you are doing other things (like hyperbolic geometry or projective geometry), then one of the other completions may be more useful.

This is a result of point-set topology. There is a one-point compactification for any locally compact Hausdorff space. I put the definitions you need (for a terse but rigorous run at it) down below as a starter for some words in case you care to go into details. Don't worry if it scares you! If you study mathematics, this will all become clear in time. I just felt like writing. This is all in the following notes; you should go there for a fully fledged proof of the result (p. 60, 3.7.1): http://folk.uio.no/rognes/kurs/mat4500h10/topology.pdf

The statement of the theorem is as follows:

One-point compactification: Let $X$ be a locally compact Hausdorff space, and $Y=X\cup\{\infty\}$. Give $Y$ the topology consisting of the open sets of $X$ together with, for each compact subset $K$ of $X$, sets of the form $Y-K$. Then $Y$ is a compact Hausdorff space. Proof is in the notes above.

Topological space: A topology on a set $X$ is a collection $T$ of subsets of $X$ called open sets such that 1) $T$ contains $X$ and the empty set $\emptyset$, 2) $T$ is closed under arbitrary unions, and 3) $T$ is closed under finite intersections. We say that a subset of $X$ is closed if its compliment is an open set (note that $X$ and $\emptyset$ are both open and closed subsets of $X$). We call the pair $(X,T)$ a topological space, and often denote it by just $X$ and think of the topology $T$ as given.

Here closed under finite intersections means if you take a finite collection of open sets, their intersection is an open set. Similarily for closure under arbitrary unions. This is very different from what we called a closed subset of $X$.

Neighbourhood: If $x\in X$ we say that a neighbourhood of $x$ is any subset of $X$ containing an open set containing $x$. (Beware that some authors define a neighbourhood of $x$ simply as an open set containing $x$; we call these open neighbourhoods.)

Hausdorff: We say that a space $X$ is Hausdorff if for each pair of distinct points $x,y$, there exist disjoint open neighbourhoods of $x$ and $y$.

Compactness: An open covering of $X$ is a collection of open subsets of $X$ such that their union is all of $X$ (note that with these definitions $T$ is the biggest possible open covering of $X$). A space $X$ is called compact if each open covering has a finite open subcovering.

Subspaces: We will want to talk about compact subsets $K$ of $X$, and we can by viewing them as subspaces of $X$ in the subspace topology. That is the set $K$ together with the topology induced by $X$; open sets of $K$ are sets of the form $U\cap K$ with $U$ open in $X$.

Compactness, as you might have guessed, is a very nice property. It is supposed to capture our intuition of being closed, small and bounded. In fact, any compact subset of a Hausdorff space is closed in the technical sense. Also, any compact subset of a metric space has a well-defined diameter. It also gives a finiteness condition that can help us out in all sorts of ways, and it has some nice formal properties: the product of an arbitrary family of compact spaces (with the product topology) is compact (Tychonoff); the image under a continuous map of a compact space, is compact.

A closed and bounded interval of the real line is compact, and (to some, more importantly) all spheres are compact!

Local compactness: We say that a space $X$ is locally compact at $x\in X$ if there is a compact neighbourhood of $x$ (that is, there is a compact subset $K\subseteq X$ and an open subset $V\subseteq K$, with $x\in V\subseteq K\subseteq X$).

There are two main types of infinity used in single-variable complex analysis. But first, let me rephrase the question. When you talk about "types of infinity" what you're really talking about is different useful compactifications for the purpose of complex analysis. The reason why one settles for the one-point compactification of the complex plane is it gives you a complex manifold -- the Riemann sphere. So it allows for convienient descriptions of things like Moebius transformations and meromorphic functions.

But other natural compactifications appear in complex analysis, the primary one being the Poincare disc. The core idea is to consider "infinities" to consist of asymptotic directions of curves "heading off to infinity". You can apply this to the complex plane itself, but it's standard to apply it to $\{ z \in \mathbb C : |z|<1 \}$, the open disc. From this perspective, "infinity" is all the points of the form $\{z \in \mathbb C : |z|=1\}$ and the Poincare disc is $\{ z \in \mathbb C : |z|\leq 1 \}$. This is a natural setting for hyperbolic geometry, if your goal is to put it into the language of complex analysis.