Why is a 100 heads run surprising? Assume we have a fair coin. We flip it 100 times. The outcome is all heads.
Why is it that all heads outcome is more surprising to us than a "more random looking" outcome with less regularity?
Aren't all outcomes of the same probability of $2^{-100}$?
And to make it a bit more statistical question, what intuition does tests like $\chi^2$ try to capture? If a sequence with a lot of regularity and a sequence with much less regularity both have the same probability, why would I distinguish between them? Why would I consider one more surprising than the other?
originally asked at MSE, but didn't get an answer that was satisfactory enough.
 A: You're right; there's nothing special in terms of likelihood about 100 heads. You're right about that. One deserves to get equally excited about 20 heads, then 3 tails, then 6 more heads, then all tails.

This leads to my slant on this, based on Bayes' rule: seeing 100 heads in a row leads us to doubt our certainty that it's a fair coin with $f=0.5$—our hypothesis $\mathcal{H}$ about the data-generating process. There are other data-generating processes that could exist, which would better explain the 100 heads.
$$
p(\mathcal{H} \mid D) = \frac{p(D \mid \mathcal{H}) \times p(\mathcal{H})}{p(D)}
$$
Even if all hypotheses are equally likely, it's more natural for these observations to come from a bent coin—or even a two-sided trick coin! That's why these surprise us. It makes us question our model of the world.  With a more even dispersion of heads and tails (a member of the 'typical set' for this distribution), it does not lend the same credence to these alternative models.

The $\chi^2$ part of the question seems unrelated.
A: Arya's answer is great.  I'll offer the frequentist take.  First off, all outcomes are not equiprobable under common assumptions.  Some sequences are equivalent under the assumption that the former flip tells you nothing about the next flip. If this is a dubious assumption, then we could talk about probabilities of sequences, but under this assumption its the number of heads that matters most.  We call this assumption "independence".
Under independence TTHHH is the same as HTHHT.  The probability of seeing a given number of heads in a sequence is a well studied distribution called the binomial.  I will leave that with you to research.
