I open a few questions in multiple browser tabs from SO and start typing answers or comments and I understandably get hit with the first eCaptcha. OK so SO thinks I might be bot because I am submitting a few entries in a short period of time. I prove I am human. From that point every time I answer or comment, the annoying recaptcha pops up again. I just proved I am human. Stackoverflow needs to remember I am not a bot and not a bot which cracked reCaptcha. I am just a fast typing human! Probably intentionally producing friction to slow me down.
The problem with your I-proved-I'm-human suggestion is that when a human solves one reCAPTCHA, they have free run of the system.
CHAPTCHAs are designed to verify and throttle suspicious activity. If you're getting reCAPTCHA'd that often, you can understand why that might be considered outside the normal use-case for the system.
If a single, human-solved CAPTCHA proves that someone is human from then on, it would open the system to a torrent of automated spam posts from that user. ReCAPTCHA simply slows down that process, making it not worth the effort for the spammer. It's a minor inconvenience under "normal use."