Are there any numbers which don't fall into at least one of the categories of real numbers, imaginary numbers, or complex numbers?
Yes, here's one:
\displaystyle \frac{1-x}{1+x^2}.
"Wait!", I hear you cry. "That's not a number, It's a function!"
OK, let's think about that for a second.
Many years ago, when the human race was younger, the only "numbers" were 1, 2, 3, 4 and so on. In fact at one point it probably was "1, 2, 3, 4, plenty" and that's it. Much more recently, when you, dear reader, were younger, you probably held similar views.
Then we added the rational numbers, like \frac{4}{5}. "Wait!", someone must have said. "That's not a number, it's a ratio! It just expresses the fact that there are 5 ellsworths in 4 vrangels!". OK, we can call them "fractions" if you prefer, who cares? We learned to add and subtract and multiply them and at some point we decided to adorn them with the term "Numbers".
Does this change them? No. Does it change us? No. Does it change the world? No. It's just a word. We call these guys "rational numbers" but we can also call them "duplexes" and nothing changes.
Then we introduced negative numbers. And real numbers. And complex numbers. We still call them numbers, although they don't enumerate anything. How many apples is -3 apples? How long is a journey of \frac{1+i}{2} miles? [1]
We call them numbers because they are useful, because we can add and subtract and multiply and divide them, and that's it. We could have called them "Splorks" and reserve the word "Number" just to the natural numbers 1, 2, 3 and so on. Wouldn't matter a bit.
There are, actually, other objects in mathematics that are usually called "Numbers". I'll review some in a minute. The thing to understand though is that this is totally arbitrary. To go back to my first example, objects like \frac{1-x}{1+x^2} are usually called "rational functions" but we could just as well have called them "hypernumbers" or "functional numbers" or whatnot and then they would get included in lists of things that are numbers. It doesn't matter. It's not an interesting observation.
What is interesting is that there are lots and lots of wondrous objects in mathematics which, like the natural numbers, can be added and subtracted and multiplied, and they tell us really deep things about mathematical worlds and about the physical world.
- There are elements of finite fields, like the "numbers" 0 and 1 which are added like 1+1=0. Do not be fooled by their commonplace appearance! Those aren't the 0 and 1 you are familiar with.
- There are Gaussian integers which look like 2+3i. Yes, they are complex numbers, but not all complex numbers are Gaussian integers, only those that have whole real and imaginary values.
- There are quaternions, sometimes called quaternion numbers, which look like 6-2i+23j-4k. They are added as you might expect, but multiplication is weird. Beyond the quaternions lie the octonions and further out the exotic sedenions.
- There are p-adic numbers, which may look like ...5141.3. Yes, there are now infinitely many digits to the left of the decimal point.
- There are cardinal numbers like 2^{\aleph_0}. They count to infinity and beyond.
- There are surreal numbers like \frac{7}{3-\sqrt{\omega}}. They include infinitesimal numbers and infinite ordinal numbers and are further generalized by amazing objects called games.
- There are rational functions like \frac{4-5x}{7+x^2}. They're not normally called numbers but, as I explained, this is just a historical accident. They form a field, meaning they allow the usual arithmetical operations to be applied to them.
- There are formal power series like 1+x+2x^2+3x^3+5x^4+\ldots. Can be added and subtracted and multiplied. Are awesome.
- There are matrices which look like \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}. They are added as you might expect, but multiplication is weird. Also not normally called numbers because they look like they are made up of several numbers, but hey, so are complex numbers, right?
As you can see, some of the things on this list are called "numbers" and some aren't, but there's no clear distinction between those that are and those that aren't so we shouldn't care about that.
Mathematics positively explodes with algebraic structures, and they are really, really beautiful. It's great to get to know them and learn how they interact and what they can teach us.
[1] Some commenters were annoyed at me for misunderstanding that a journey of \frac{1+i}{2} miles has length \sqrt{2}/2. Nope, that’s a journey from 0 to the point \frac{1+i}{2}, which isn’t what I said. A journey of length \sqrt{2}/2 has length \sqrt{2}/2. A journey of “length” \frac{1+i}{2} doesn’t exist because non-real numbers aren’t measuring the length of anything.