Drawing enough samples from a Normal distribution leads to a distribution approximation very close to the Normal distribution.

set.seed(4499864)
i20trip <- FALSE
for (i in c(seq(1,20,by=0.5),rep(20,5))) {
  n <- floor(2^i)
  if (!i20trip) {
    x <- rnorm(n)
    i20trip <- i == 20
  }
  hist(x,freq=FALSE,ylim=c(0,0.5),xlim=c(-4,4),breaks=c(-Inf,seq(-4,4,by=0.1),Inf)
       ,main=sprintf("Sample = %7d",n),col='indianred',border='indianred')
}

In addition, looking at the Quantile-Quantile (QQ) plot comparing the sample to the Normal is informative. Note how the tails creep out as the sample increases. If these data were test statistics, then this implies that as the number of tests increaeses, even under the null hypothises, more extreme \(p\)-values will occur if enough tests are performed. This is the multiple hypothesis testing problem.

i20trip <- FALSE
for (i in c(seq(1,20,by=0.5),rep(20,5))) {
  n <- floor(2^i)
    if (!i20trip) {
    x <- rnorm(n)
    i20trip <- i == 20
  }
  qqnorm(x,xlim=c(-4.5,4.5),ylim=c(-4.5,4.5),col='dodgerblue')
  qqline(x)
}