No one quite knows, as no such system exists yet, though I imagine SpaceX has some internal numbers. You need enough to slow your mostly-empty (and thus fairly light) stage from its terminal velocity to 0, and however many seconds margin you need to be comfortable.

Let's do some very rough numbers to get a ballpark...

Say terminal velocity is 100 m/s, which is fairly fast, but a rocket is a skinny streamlined thing. (It could fall slower, maybe even 50 m/s, on its side or with parachutes, but parachutes are heavy and complicated, and falling sideways may be difficult to control, but seems quite possible.)

Say an F9 "1.1" first stage weighs 28,000 kg. The landing components mostly are already there: the engines are already gimballed and the additional software weighs nothing. Unless you're so awesome you can land in a cradle--which is not impossible--you need some landing gear, so let's add 3000kg, for a dry weight of 31,000 kg.

Amongst all its engines, our F9 has some 300,000kg of thrust. Just one engine is around 33,000kg thrust, so we'll probably only use one or two.

Two engines will get us easily about 10m/s slower per second, so we need to retro-fire for minimum 10 seconds. You probably want some margin over that; let's reserve 20 seconds fuel.

F9 1.1 stage 1 will have a burn time of 185s; our retro burn is only at about 25% capacity, so 20s of two engines is worth, oh, 5s of normal burn time. Say about 3% of the fuel.

Of course, it's not that simple, because (1) we've made the stage heavier with the landing gear, which requires more fuel and (2) to avoid losing performance we'll want to add enough fuel to add our extra burn time, which also requires more fuel, and (3) we need more fuel to lift all that fuel and mass we added. The rocket equation strikes again!

(Of course, it may not be that bad, because usually first stages have plenty of margin, and most payloads are not max. Maybe if you encounter a really heavy payload you just agree to throw away the rocket that time; use an old one or something.)

Most of my assumptions are pessimistic (I think) but in the real world they may turn out optimistic. I would guess you're looking at 3-5% of the fuel load for a recoverable VTOL stage. A lot depends on details, as with rockets weight scales very badly against performance.

Oh, and I should also point out that Armadillo Aerospace originally considered rocket-only recovery. I don't know if they posted any numbers on the necessary fuel reserves; I did look briefly but didn't see anything, and don't remember any specific numbers. May or may not be any published.

They have lately been working on parachute-only recovery, apparently having decided that the mass/complexity penalty of parachutes was better than that of retro-fire fuel. They've lost several vehicles already to minor parachute problems. Parachutes aren't easy.

SpaceX, working at a different scale (though not a different flight regime, at least for the first stage) still thinks retro-rocket landing to be the answer. I imagine an Armadillo-size rocket is less likely to have its various parts crushed at a parachute-speed descent than an F9 stage. Armadillo does occasionally get a STIG back in one piece, and parachute-into-water F9 stages are known to be completely trashed. Dry weight would seem to make a difference here.