Likelihood of rejecting a fair coin (repeated significance testing) Suppose I have a fair coin and I flip it numerous times, testing after every time using Pearson's $\chi^2$ test of fit to fairness. What is the likelihood that I will, at some point, reject that the coin is fair (for given $\alpha$)? If (as I suspect) that's $1$, then is there an expected number of flips after which I'll reject that the coin is fair?

(This precise example comes up when trying to explain to coworkers by analogy why one can't repeatedly peek at split-test results without accounting for the repeated testing.)
 A: This is a problem involving what are essentially "ruin probabilities" of a stochastic process.  The fact that the tests go on forever is not sufficient to imply that the probability of eventual rejection is one.  You would need to establish this formally via analysis of the ruin probability.  It is also notable that the tests are not independent, since the test statistic at higher $n$ values is related to the test statistic at lower $n$ values.  I will set up the problem for you below, and give you an outline of how you can prove your conjecture.

Setting up the problem: You have a sequence $X_1, X_2, X_3 \sim \text{IID Bern}(\theta)$ and your null hypothesis of a fair coin is $H_0: \theta = \tfrac{1}{2}$.  After $n$ tosses the expected number of positive indicators is $\mathbb{E}(n \bar{X}) = \tfrac{1}{2} n$, so the Pearson test statistic (under the null hypothesis) is:
$$\begin{equation} \begin{aligned}
T(\mathbf{x}_n) 
&= n \Bigg[ \frac{(\bar{x}_n - \tfrac{1}{2})^2}{\tfrac{1}{2}} + \frac{(1-\bar{x}_n - \tfrac{1}{2})^2}{\tfrac{1}{2}} \Bigg] \\[6pt]
&= n \Bigg[ \frac{(\bar{x}_n - \tfrac{1}{2})^2}{\tfrac{1}{2}} + \frac{(\tfrac{1}{2}-\bar{x}_n)^2}{\tfrac{1}{2}} \Bigg] \\[6pt]
&= 4 n (\bar{x}_n - \tfrac{1}{2})^2. \\[6pt]
\end{aligned} \end{equation}$$
Higher values of the test statistic are more conducive to the alternative hypothesis, and for large $n$ we have the asymptotic null distribution $T(\mathbf{X}_n) \sim \text{ChiSq}(1)$.  We define the critical point $\chi_{1,\alpha}^2$ by:
$$\alpha = \int \limits_{\chi_{1,\alpha}^2}^\infty \text{ChiSq}(r|1) \ dr.$$
Then, assuming you use the chi-squared approximation for your p-value (rather than the exact distribution of the test statistic) we have the rejection region:
$$\text{Reject } H_0 \quad \quad \quad \iff \quad \quad \quad 4n (\bar{x}_n - \tfrac{1}{2})^2 > \chi_{1, \alpha}^2.$$
Hence, the "ruin probability" you are looking for is:
$$W(\alpha) \equiv \mathbb{P} \Big( (\exists n \in \mathbb{N}): 4n (\bar{X}_n - \tfrac{1}{2})^2 > \chi_{1, \alpha}^2 \Big).$$

Establishing the result: You have conjectured that $W(\alpha) = 1$ for all $0 < \alpha < 1$.  The rejection events in your sequence are not independent, which makes the problem tricky.  Although the rejection events are not independent, it should be possible to form an infinite series of events which each have a positive lower bound on the probability of rejection, regardless of the previous data.  We do this by splitting into subsequences and conditioning on a sample mean of one-half in the previous data.
To prove your conjecture, define a sequence of positive integer values $m_1^*, m_2^*, m_3^*, ...$ and then divide the main sequence of tosses into disjoint finite subsequences of these lengths (i.e., we divide an infinite sequence into an infinite number of finite subsequences).  Let $\bar{X}_k^*$ be the sample mean corresponding to the $k$th subsequence, and note that these sample means are independent, since the subsequences are disjoint.  
Define the test statistics up to the ends of these subsequences as:
$$T_k \equiv T(\mathbf{x}_{n_k}) = 4 n_k \Bigg( \frac{1}{n_k} \sum_{i=1}^{n_k} x_i - \frac{1}{2} \Bigg)^2 \quad \quad \quad \quad n_k \equiv \sum_{i=1}^k m_i^*.$$
With a little algebra we can show that:
$$T_k = 0 \quad \quad \quad \implies \quad \quad \quad T_{k+1} = 4 m_{k+1}^* (\bar{x}_{k+1}^* - \tfrac{1}{2})^2 \times \frac{m_{k+1}^*}{n_k + m_{k+1}^*}.$$
Under this condition we have:
$$\text{Reject } H_0 \quad \quad \quad \iff \quad \quad \quad 4 m_{k+1}^* (\bar{x}_{k+1}^* - \tfrac{1}{2})^2 > \frac{n_k + m_{k+1}^*}{m_{k+1}^*} \cdot \chi_{1, \alpha}^2.$$
Now, choose some value $0 < \alpha^* < \alpha$ so that $\chi_{1, \alpha}^2 < \chi_{1, \alpha^*}^2$.  Choose each value $m_k \in \mathbb{N}$ sufficiently large so that you have:
$$\frac{n_k + m_{k+1}^*}{m_{k+1}^*} \cdot \chi_{1, \alpha}^2 \leqslant \chi_{1, \alpha^*}^2 \quad \quad \text{for all } k = 0,1,2,....$$
The condition $4 m_{k+1}^* (\bar{x}_{k+1}^* - \tfrac{1}{2})^2 > \chi_{1, \alpha^*}^2$ is sufficient to ensure rejection in the $(k+1)$th subsequence, even with the lowest possible evidence for rejection in the previous data.  This establishes that the probability of rejection in any one of the subsequences is at least $\alpha^*$, regardless of the presvious data.  Hence, we have:
$$W(\alpha) \geqslant 1 - \prod_{i=1}^\infty (1-\alpha^*) = 1-0 = 1.$$
A: For one test with a given $\alpha$, you have a $p$ value and a confidence level $q=1-p$.
Now, if you do $n$ independent test all with the same $p$ value, your confidence $q^n \rightarrow 0$ as $n$ increases. So you are almost sure to fail the test if you repeat independent tests for a large enough $n$. 
Now a $\chi^2$ test for $n+1$ coin toss is far from being independent from the test for the $n$ first tosses, so that the convergence of confidence to 0 has to be much slower than $q^n$. However, the first $100^{n+1}$ tosses are almost independent of the first $100^n$.
So that a test with a fixed $p$-value repeated for arbitrarily large $n$ is almost sure to fail.
For a small value of $n$ however, the successive pearson tests suggested will not be independent, but it is intuitive that their $p$-value is strictly lower than implied by $\alpha$.
