Probably, but I think even in a frictionless system you could still get the same effect.
I suspect even if slip between teeth, slack in system, etc... was eliminated you could still get the same effect.
The amount of energy in the system isn't the same thing as the amount of movement, so in a perfectly efficient system (I think) the initial gear would spin faster than the final gear but the final gear would turn with more force (same amount of energy).
Friction basically. In a perfectly idealised version of the system with no friction/slack/materials being deformed/etc, if you started it off then let go then everything would keep spinning forever. You don't need to pump a whole lot of energy into the device to make it go.
That's a big if. I haven't done the math yet, but I'll bet that the outside rim of at least one of the gears would have to be started moving faster than the speed of light. I'm not sure how you would do that.
The gear just before the first gear at light speed would have the outside rim moving at about 100th of the speed of light. It would take quite a bit of energy to do that.
(Others have pointed out that at speeds well before the speed of light, some of the gears would melt. I'm ignoring that.)
The energy not lost to friction is being loaded into the system as the gears and axles start to strain from the stresses applied. Some energy is potential energy like a compressed spring. The balance is spent making the material yield; this will be lost as heat.
