Computable Functions are Rare
When stating mathematical problems, we usually state them in terms of elementary functions, but most certainly computable functions, because those are the only ones we know how to write down in finite space!
Because our brains can only explicitly conceptualize the computable functions, we have an innate bias towards thinking about these functions, and giving them a centrality within the world of numbers. When you read about diagonalization, it is tempting to think: "Non-computable numbers are such a hassle to build! Surely they must be rare!" But the reality is that the computable functions are the infinitely endangered species! There are only $\aleph_0$ such functions, but at least $\mathfrak c$ non-computable functions.
There are many ways to go from a computable number/function to a non-computable one (diagonalization being a well-known example, and the Halting Problem being another), but I would be surprised if anyone can name a "natural" problem which starts with a non-computable function/number and whose solution is computable (and by "natural", I mean one that isn't specifically contrived to do this).
An equation defines the intersection of two functions. If you take two arbitrary functions, what are the odds that those functions will intersect on one of the infinitesimally probable countable functions? This is why mathematicians are surprised when a result has a nice closed form. Usually, only trivial problems have this property.
Names Won't Save You
Of course, there is this business of deciding what is an "elementary function" or an "analytical solution". The answer is: "It doesn't matter." Those questions are totally irrelevant. Pick any finite set of problems that you like. Let us assign names to the solutions of those problems, regardless of whether they are computable or not. Now, we have greatly expanded the realm of "elementary functions". Awesome!!! We even did something amazing...we added some non-computable functions, which should really ramp-up our problem-solving power, right? Well, unless you got extraordinarily lucky, I'd bet against it.
An arbitrary non-computable function is garbage. It's less than worthless. While it is a solution to infinitely many problems, and thus expands your ability to write "closed form solutions" by a factor of at least $\aleph_0$, I'd wager, it is also not the solution (or remotely relevant) to a larger infinity of problems. Those problems require different non-computable functions than the ones you named.
Ok, ok...I'll let you cheat. I'll let you open up the toolbox and add some more functions. I didn't say how many you could add before, only that they had to be finite in number. You could have added a googol functions, I don't care. This time, I'm going to be really generous. I'm going to let you add an infinite number of non-computable functions, up to $\aleph_0$ of them!
Surely now we can write down nice "algebraic" solutions for most problems, given that we have beefed up our toolbox by a factor of infinity! But sadly, no. Our infinity isn't nearly big enough. No matter how clever you were at picking non-computable functions, there will still be infinitely many problems whose solution requires a non-computable function that you didn't choose.
You know what? I'm feeling generous. I feel bad, because you want mathematics to be nice and beautiful, and so far, it just looks like a giant mess. We tried to impose order by naming lots of solutions that didn't have names before. And as long as we stay under the $\aleph_0$ threshold, we can assign finite names to our "Augmented Algebraic Functions Toolbox". I'm going to do you one last favor. I'm going to let you add as many non-computable functions as you want! That should fix this problem once and for all, right?
Well, no. Now we have just traded one problem for another. If we simply add all functions $f: \mathbb R \rightarrow \mathbb R$ to our toolbox, we indeed capture a truly mind-bending number of functions, including more non-computable functions than you can shake a stick at! But the problem now is that we cannot name them! I mean, we can name them. We can put them in one-to-one correspondence with the reals. But, unfortunately, that means we cannot write most of them down! The only ones we can write down are the ones with a finite representation...and there's only $\aleph_0$ of those... D'oh!
And if we're being practical, nobody is going to read a paper which uses "elementary functions" with names that are 100 characters long. Probably 100 "elementary functions" is pushing the patience of most mathematicians. Unfortunately, $100 \lll \aleph_0 \lll \mathfrak c$. And so it goes...