We're looking for a way to register a user without requiring that they specify a password and instead mail them the password as part of the account verification process.
Is this even remotely possible with EE or a module like Expresso FreeMember?
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FWIW Simple Registration has support for disabling passwords in the signup form, auto-generating a password and emailing that password to the user.
On your form, you could set the password field as "hidden" and give it a value. (Either hardcode in a value or - better yet - have some randomly-generated string as that field value.) Then, as part of the account verification process, they're shown a link to retrieve their "lost" password and reset it.
The downside there is that the password - whether randomly-generated or hardcoded - would be visible in the page source. This could net you a lot of spam signups, which is never fun.
Is there a reason you're doing this? If you're trying to cut down on spam signups, and force a multi-step account verification procedure, couldn't you route everyone into a certain member group, and have all new members perform some minor task (entering some data into a form that's visible only to members in that group, for example) - and then everyone who doesn't perform that task in five days, their membership is either deleted, or they're moved to the Banned members group?