Adaptively selecting the number of bootstrap replicates As with most Monte Carlo methods, the rule for bootstrapping is that the larger the number of replicates, the lower the Monte Carlo error. But there are diminishing returns, so it doesn't make sense to run as many replicates as you possibly can.
Suppose you want to ensure that your estimate $\hat θ$ of a certain quantity $θ$ is within $ε$ of the estimate $\tilde θ$ that you would get with infinitely many replicates. For example, you might want to be reasonably sure that the first two decimal places of $\hat θ$ are not wrong due to Monte Carlo error, in which case $ε = .005$. Is there an adaptive procedure you can use in which you keep generating bootstrap replicates, checking $\hat θ$, and stopping according to a rule such that, say, $|\hat θ - \tilde θ| < ε$ with 95% confidence?
N.B. While the existing answers are helpful, I'd still like to see a scheme to control the probability that $|\hat θ - \tilde θ| < ε$.
 A: If the estimation of $\theta$ on the replicates are normally distributed I guess you can estimate the error $\hat{\sigma}$ on $\hat{\theta}$ from the standard deviation $\sigma$:
$$
\hat{\sigma} = \frac{\sigma}{\sqrt{n}}
$$
then you can just stop when $1.96*\hat{\sigma} < \epsilon$.
Or have I misunderstood the question? Or do you want an answer without assuming normality and in presence of significant autocorrelations? 
A: On pages 113-114 of the first edition of my book Bootstrap Methods: A Practitioner's Guide Wiley (1999) I discuss methods for determining how many bootstrap replications to take when using the Monte Carlo approximation.
I go into detail about a procedure due to Hall that was described in his book The Bootstrap and Edgeworth Expansion, Springer-Verlag (1992). He shows that when the sample size n is large and the number of bootstrap replications B is large the variance of the bootstrap estimate is C/B where C is an unknown constant that does not depend on n or B. So if you can determine C or bound it above you can determine a value for B that makes the error of the estimate smaller than the $\epsilon$ that you specify in your question.
I describe a situation where C = 1/4. But if You don't have a good idea as to what the value C is you can resort to the approach you describe where you take B=500 say and then double it to 1000 and compare the difference in those bootstrap estimates. This procedure can be repeated until the difference is as small as you want it to be.
Another idea is given by Efron in the article "Better bootstrap confidence intervals (with discussion)", (1987) Journal of the American Statistical Association Vol. 82 pp 171-200.
