In all likelihood, you have a poorly diagnosed complete separation / perfect prediction in your model: a combination of the explanatory variables (if you used interactions), or more likely a single explanatory variable, uniquely identifies one of the rare events. Let's say that if x > 10, then the outcome is always a one, while for x < 10, there can be a mix of zeroes and ones. What happens then is that the greater the coefficient for x, the closer you can get the predicted probability to 1 for the cases with x > 10. Since their contribution to the likelihod is $\ln \hat p_i$, maximum likelihood keeps pushing that number up to the extent possible (while maintaining the other coefficients in bay so that the probabilities for x<10 are OK), and sky is the limit... except that the finite precision of computer arithmetic prevents that from technically happening, so you will stop somewhere around $\hat p_i = 1-10^{-8}$ or so. This is a known problem for glm in R; Stata diagnoses this and drops the perfectly predicted observations.
You need to identify which of your explanatory variables perfectly predicts the outcome, and do something about it -- exclude it from regression, find another measure of the underlying concept, etc. Another solution is to use Firth logistic regression, which is a frequentist version of Bayesian regression with Jeffrey's prior, or, in a distant way, a version of ridge regression for binary outcome.
