Don't know if this really is what you're looking for, but I'll throw it out anyway.
First we start with the sponsor debt mechanics (ie, what admiralducksauce suggested). This gives you a solid core for asking for little bits of help -- for a price.
But take it a step further. Allow the character to make extensive use of the "Temporary Powers" sidebar on YS92, but using debt rather than Fate points. Work with the player to make a list of powers that are appropriate to borrow from a demon. This should certainly include the physical powers (Strength/Toughness/etc), and might well include a number of others (Cloak of Darkness, Hellfire, and more). Basically, for a future favor (or favors), the character can access -- for a scene -- as many of these powers as he's willing to pay the price for.
And I'd consider adding just one more small caveat, if you think the player would be cool with it. Pick a reasonable number in advance (perhaps trouble comes in threes?). Once the character has "borrowed" a particular power that many times, it sticks. He adds it to his character sheet permanently, reducing refresh and losing Fate points as though he'd just bought the power. He's just lost a piece of his soul, and become a little more like the demon he's dealt with.
In effect, the result would be a somewhat Changeling-esque character, who struggles between the power on the offer and the desire to retain his soul. Of course, if his refresh reaches zero, the Choice is made.