Chaos and the Logistic Equation

It was in a class last fall in which I first learned a mathematical definition of chaos, and an example that exhibits chaotic behavior.  That example was the logistic equation.  Put simply, to get the next number xn+1 from the last number xn, you use the formula xn+1= r*xn*(xn-1).  That didn't look that crazy to me.  But apparently when r is greater than 3.57, you get chaotic behavior.  Not only that, but they say there's no explicit formula, in other words, if you have some value for r and some starting value x0, if you want the 100th value in the sequence you have to compute all 99 others in between, there's no formula where you plug in your 100 and just get the 100th value.  But it seemed so simple!  I knew I had experience with similar problems, recalling my efforts in finding a formula for the Fibonacci sequence.  So, I had to see what I could do with this problem.

I started off the only way I knew how: simply.  Say xn+1=2*xn.  In that case, xn=x0*2^n.  The extension to xn+1=a*xn, was simple enough, as xn=x0*a^n.  All right, what about xn+1=xn+a?  Easy, xn=a*n+x0.  I worked my way up to a formula for xn+1=a*xn+b (which I don't know off the top of my head and don't have the paper I wrote it down on).   Then I was ready for the next level.
What about xn+1=xn^2?  Well, I could solve that.  It was something like x0^(2^x), and now I had a foothold on this level.  I was looking for a solution to xn+1=a*xn^2 +b*xn+c.  All I had to do was find the correct additions and extensions to the formula and I would solve that problem along with the logistic equation.

I got farther.  I got to what I believed was "close."  But I couldn't solve it.  I guess there was a reason it was proven to be impossible.
It was proven to be impossible under certain conditions, in a certain realm.  But that didn't change the fact that it existed.  Even if it couldn't be expressed with basic operations, maybe, with something new, some other function, some new expansion in the world of mathematics, you could come up with a solution.
I still feel that way to some extent now, looking back at the problem.  Maybe, if I were creative enough, clever enough, I could find an answer...