Man, there's some bad advise here. "I'm very eager to start, and can accept the offer immediately for $X/hr, but if this is the best you can do I'll need Y day

Man, there's some bad advise here. "I'm very eager to start, and can accept the offer immediately for $X/hr, but if this is the best you can do I'll need Y days to get back to you." If you phrase it this way, there's basically zero chance they'll revoke the offer they've already extended - it'll just make them a tiny bit worried they've lowballed you and think you might be entertaining other options.