(without the homebrew I'm dealing with....)
Maximized Time Stop gives 4+1=5 rounds for the caster to do things on his own turn.
Extended Time Stop gives 2*(1d4+1) rounds for the caster to do things on his own turn.
Extended Maximized Time Stop gives 2*(4+1)=10 rounds for the caster to do things on his own turn.
Extended Empowered Maximized Time Stop gives 2*1.5*(4+1)=15 rounds for the caster to do things on his own turn.
Extended Intensified Time Stop gives 2*2*(4+1)=20 rounds for the caster to do things on his own turn.
Extended Intensified Time Stop doesn't make the caster feel like 20 rounds are passing, but only 1d4+1 rounds of actual actions can be taken. That's ridiculous.
Persist Spell works exactly like all the other metamagic effects - the caster gains that many rounds of actions during his own turn. With Persist Spell, it's a whole 24 hours worth of rounds of actions.
Yes, that's wildly broken, but it's completely RAW. If you want to nix it, you're going to have to just out and out Homebrew rule that Time Stop can't be Persisted.
Nothing wrong with that, but don't try to twist things to somehow say that it's RAW that Time Stop can't be Persisted.