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What is the simplest or easiest or most clear way that a mathematician could discover the normal distribution and the central limit theorem?

The derivation does not need to be rigorous. (Much of calculus was discovered and understood clearly before rigorous proofs were provided, and maybe a similar thing is true for the central limit theorem.)

It's ok if the answer is not historically accurate. (But I'd also be interested in knowing how the normal distribution was discovered historically.)

Is there a viewpoint that makes the central limit theorem intuitive or "obvious"?

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    Analysis does. The probability density of the sum of two random variables are the convolution of their individual probability density functions and a convolution is as least as "nice" as the nicest of the involved factors. – 2017-02-20
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    One way to discover the normal distribution is by assuming that the least square estimation and the MLE estimation of a linear regression is the same. I believe this is approximately how Gauss discover the normal distribution. – 2017-02-20
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    @mathreadler Could you elaborate on that? How does that lead to the normal distribution? – 2017-02-20
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    @RanWang I'd be interested in hearing more details about your comment also. – 2017-02-20
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    @eternalGoldenBraid I will be looking for the exact history when I have time...... for now, I am sorry but you have to wait a little bit...... – 2017-02-20

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Normal distribution discussion:

For centuries the sciences and mathematics wanted to 'get it'.

Consider first the contrast of the Laplace distribution with the Standard Normal Distribution:enter image description here.

Laplace took a stab at it but of course it left something to be desired.

You are looking for a pdf that has more 'integrity' than Laplace's 'splice job'. You think about the amazing symmetry and beauty of the parabola. And how interesting that $y = x^2$ 'flattens numbers' less than $1$, but 'bumps em up' when larger. You have an idea - compose $f(x) = x^2$ with $g(x) = e^{-x}$ - $g f(x))$ sure has smooth symmetry!

As you work out the details to create a pdf, you are tickled pink when you discover that one-standard deviation corresponds to plus/minus 1 and that the inflection points also reside there - your continuous distribution gets really serious flattening stuff outside of $[-1, +1]$.

Of course you still have a lot to learn about this creation, but you actually did hit the jackpot.