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You might be able to build a friction drive e-bike for $300.
Friction drive meaning you put a tube on the driveshaft of the motor and push it against the tire to move the bike. With gas motors, people typically use bmx bike pegs (the ones that you see attached to the wheel shaft of small bikes) and mount the pegs to the motor shaft. You can also use epoxy mixed with sand on any shaft to provide friction. Friction drives don't work as well in rain.
You would have to figure out the gear ratio on an electric motor. The motor would have to be held in a frame that pivots and can be held down against the tire, or can be lifted up for free pedaling with no motor drag.
You might find a 1 kW motor for $50. Hopefully you have a lot of junk in your workshop that can be transformed into parts for a frame and driveshaft. 48V and 12 amp hours of lead acid will run you about $150. A 48V charger another $150, but you can also disconnect things and run multiple cheap 12V chargers if you have them available (this is a pain). Wiring needs to be 12 gauge (minimum) or 10 gauge (preferred), hopefully you have this in your junk drawer.
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