Why Normal distribution is so special? There are bunch of symmetric distributions with mean zero and thin tails. Why this is the fact that in the CLT the limiting distribution is Normal? What is so special about it, why the nature "has chosen" Normal to be the limiting distribution in a lot of processes?
 A: A Normal distribution arises naturally in any system where you have a 'target' value plus random noise that bumps the values of the samples additively away from the 'target'.
For example, you're measuring something tiny, let's say the width of a toothpick, with an elastic tape measure. Your hands are never perfectly still - they oscillate randomly to both sides with equal probability per millisecond. Let's say - by 1 mm. 
This means that over 10 ms, they may have gone: LRLRLRRLRL (5L/5R), or LLRRLLLRLL (7L, 3R) or any other combination of Ls and Rs.
In the first case, the 'lefts' and 'rights' balance out, so at the end your hand is exactly at the end of the toothpick, so your measurement is perfectly accurate. In the second, you have a net drift of 4 mm to the left, so your measurement will be -4 mm off from the true width. 
You don't, and can't, know what the exact scenario was, so you have to take them all into account when reporting your measured width. This is a made-up scenario, but it actually comes up every day when you're working with precise measurements.  - it's not just you, it's also the temperature and pressure fluctuations, random coworkers leaning on the bench, and a billion other nuisances throwing it off a tiny bit.
Now, if you swap the Ls and Rs for 0s and 1s, this looks a lot like a Poisson distribution. Except if you want to find the drift, it's easier to model them as two Poissons, spliced back to back.
Except, again, your butterfingers are more erratic - the 1 mm is the average drift, it could be more, could be less, each time. We need an equivalent that works over reals, and that happens to be the Exponential distribution.
If you model the average drift D from the 'true' value V by a pair of exponentially-distributed 'bumps' pulling against each other, you get a Normal(V, D). Yay?
CLT arises because as you 'zoom out' from some true unimodal-ish value, the peak stays the same, but the bumps get tinier and tinier until they start looking like little exponentially-distributed fractional bumps added to the mean.
