Multiple comparisons (e.g. Bonferroni). The more experiments I do the less significant my results If I run one experiment, and obtain a p-value of $p=.03$ and I have a priori decided that my $\alpha=.05$. I have a significant a result, I conclude that my data reject $H0$.
If I run two identical experiments with the same study $\alpha$, both with a p-value of $p=.03$. Now as I have done two different tests, I correct for multiple comparisons (e.g. Bonferroni), and the $p$ the values become non-significant, as my alpha is divided by the number of tests done.
Imagine the same outcome but for hundred experiments in the same study, testing the same hypothesis.
How can it be that gathering evidence that converges in rejecting the null, makes it less significant and therefore prevents from rejecting it?
I understand that common sense (and for example a Bayesian approach), should find more evidence in rejecting the null. But, I'm more interested in knowing, if my reasoning is flawed (e.g. multiple comparisons should not be applied in this case, but then I'm interested in why) or this is a known flaw, and we just live with it, and the decision is taken based on common sense and not on a strictly mathematical proof.
 A: You seem to be confusing the requirement of replication (which reduces Type I error rates) with the problem of multiple comparisons (which increases Type I error rates).
REQUIRING REPLICATION:
If you decide a priori to make significance require two independent tests to come out p < .05, you are actually REDUCING the probability of Type I error rather than INFLATING it, because you are making the criteria for significance more stringent. Thus, this is NOT a situation that calls for Bonferroni adjustment. In fact, the probability of getting p < .05 under the null hypothesis in both of two independent studies is .05^2 = .0025.
MULTIPLE COMPARISONS:
On the other hand, if you decide that significance requires only ONE of two independent tests to come out p < .05, then you have given yourself an extra chance of getting significance, so you are INFLATING the probability of Type I error. The probability of getting p < .05 in at least one of two independent tests is 1-(1-.05)^2 = .0975.
These same principles apply if you conduct 100 experiments. If you require all of them to produce p < .05, then the probability of Type I error under the null hypothesis is reduced to .05^100, which is near zero. If you require only 1 of them to produce p < .05, then the probability of Type I error under the null hypothesis is inflated to 1-(1-.05)^100, which is roughly 99%.
A: I'm not sure that you actually ran two (or one hundred) experiments here. 
Instead, it sounds like you ran a single experiment, testing a single hypothesis, twice. As such, your "family" of hypotheses hasn't gotten larger and there is no need to correct your $\alpha$ to maintain the same familywise error rate. 
Realistically, you would be better off combining both experiments' data into a single analysis. For example, you might add a batch/session/run factor to your ANOVA or GLM, along with the experimental condition(s) of interest. This would allow you to identify changes from session to session and determine whether they interact with the treatments you applied. 
