What fraction of repeat experiments will have an effect size within the 95% confidence interval of the first experiment? Let's stick to an ideal situation with random sampling, Gaussian populations, equal variances, no P-hacking, etc. 
Step 1. You run an experiment say comparing two sample means, and compute a 95% confidence interval for the difference between the two population means. 
Step 2. You run many more experiments (thousands). The difference between means will vary from experiment to experiment due to random sampling. 
Question: What fraction of the difference between means from the collection of experiments in step 2 will lie within the confidence interval of step 1? 
That can't be answered. It all depends on what happened in step 1. If that step 1 experiment was very atypical, the answer to the question might be very low. 
So imagine that both steps are repeated many times (with step 2 repeated many more times). Now it ought to be possible, I would think, to come up with an expectation for what fraction of repeat experiments, on average, have an effect size within the 95% confidence interval of the first experiment. 
It seems that the answer to these questions need to be understood to evaluate reproducibility of studies, a very hot area now. 
 A: [Edited to fix the bug WHuber pointed out.]
I altered @Whuber's R code to use the t distribution, and plot coverage as a function of sample size. The results are below. At high sample size, the results match WHuber's of course. 

And here is the adapted R code, run twice with alpha set to either 0.01 or 0.05.
sigma <- 2 
mu <- -4
alpha <- 0.01
n.sim <- 1e5
#
# Compute the multiplier.

for (n in c(3,5,7,10,15,20,30,50,100,250,500,1000))
{
   T <- qt(alpha/2, df=n-1)     
# Draw the first sample and compute the CI as [l.1, u.1].
#
x.1 <- matrix(rnorm(n*n.sim, mu, sigma), nrow=n)
x.1.bar <- colMeans(x.1)
s.1 <- apply(x.1, 2, sd)
l.1 <- x.1.bar + T * s.1 / sqrt(n)
u.1 <- x.1.bar - T * s.1 / sqrt(n)
#
# Draw the second sample and compute the mean as x.2.
#
x.2 <- colMeans(matrix(rnorm(n*n.sim, mu, sigma), nrow=n))
#
# Compare the second sample means to the CIs.
#
covers <- l.1 <= x.2 & x.2 <= u.1
#
Coverage=mean(covers)

print (Coverage)

}

And here is the GraphPad Prism file that made the graph.
