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I have the old Google Apps Free edition and I apparently can't use Inbox, so there you go.

Same - the help page says it's not available at all for Google Apps accounts, only for personal Gmail accounts.

https://support.google.com/a/answer/6082718?rd=2

Since I only use my @gmail.com account for spam and have my own domain for personal emails, guess this is a no go for me. I was considering switching back to a personally hosted mailserver for privacy reasons anyway, would have liked to try this though.

This is quickly becoming SOP... I'm still waiting for Google Now Gmail cards >:/

(And as a single-user apps account I don't even get the recently-released unlimited storage...)

Google Apps free edition (up to 10x users) no longer exists unless you were grandfathered in. Now it starts at $5/user/month.