The upshot, though, is that while the plaintiff may find out, the Federal Rules of Evidence (specifically, Rule 411) generally keeps knowledge of the existence, let alone the limits, of any insurance policy away from the jury. See the rule, and the Notes from the Advisory Committee (which wrote the rules), at
https://www.law.cornell.edu/rules/fre/rule_411. Keeping that knowledge away from the jury avoids the temptation for them to rule against the defendant on the (impermissible) grounds that "well, he can afford it--he has insurance."