Gearhead Svellsongur, does it give extra might per gear or not?

No, Svellsongur does not give extra Might per gear on Gearhead. Here's why:

Svellsongur copies Gearhead's printed ability ("Each Equipment attached to me gives double its base Might bonus") onto itself. This creates two instances of the same doubling effect on Gearhead — but they don't stack multiplicatively. Both look at the same base values and apply the same doubling, so the result is the same as having just one copy (FAQ #8493).

Additionally, Svellsongur has a base Might bonus of +0, so doubling it contributes nothing (FAQ #8493).

Example: Gearhead (base 3 Might) with one Doran's Shield (+1) ends at 5 Might — whether Svellsongur is attached or not. With two +1 Equipment (say Doran's Shield and another), you'd get 3 + 1×2 + 1×2 = 7 Might — but that's still just Gearhead's own doubling applying to each gear once. Svellsongur just duplicates Gearhead's already-present effect, not amplifying it further.