Specifically, it does have an "override sites that don't let you remember passwords" feature - and if a site isn't let you store cookies that store credentials, LastPass can auto login when it sees the site's login page. For your requirement, just disable all timeouts in its config - for most people I'd recommend a suitable inactivity timeout.
It's free as in beer (except on mobile phones where they charge a yearly fee) but not open source. See https://lastpass.com/
KeePass is also good and open source, with many plugins and great features, but doesn't have the browser integration.
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