First, build 100km-tall tower. 100 kilometres is the official threshold of space and that's going to be our baseline.
(Of course, no tower can be just 100 kilometres tall. A real space elevator would have to go all the way to geostationary orbit, which is 38,000 kilometres up, and further beyond, so that it would stay upright due to tension in the structure rather than compression. But whatever.)
Next, connect a pedal-powered elevator car to your tower, climb in at the bottom and begin pedalling. Gears and chains convert your exerted kinetic energy into gravitational potential energy and then into height.
Meanwhile, the amount of potential energy that same person has while stationary 100km up in the air (let r = 100km), would be -GMm/(R + r) joules, or -GM/(R + r) = -61.5 MJ/kg.
The potential difference between these two points is 964 kJ/kg.
A top cyclist (this is the least accurate figure in the calculation) can put out something like 3 W/kg.
Therefore, it would take our cyclist (964 kJ/kg) / (3 W/kg) = 321,618 seconds, or approximately 90 hours of continuous pedalling, to reach space.
Assuming that our cycle platform can be locked into position when the cyclist needs to rest, and s/he can cycle for 6 hours per day, that's 15 days.
This all assumes the mass of the cyclist is constant on the ascent: all the food s/he eats is delivered to the lift car by a third party, and likewise all waste is removed. If we require our cyclist to make the ascent solo, the cycle elevator has to have all the required food onboard before departure, and the platform functions quite a lot like a conventional rocket in turns of its acceleration. The equations become a lot more complicated, however.
We're also assuming a cycling mechanism with no mass whatsoever. This is not entirely outrageous, however. Racing bicycles are already incredibly lightweight, and remember that we don't need wheels - just a seat, pedals, a gearing mechanism interlocking with a toothed rack running up the side of the space elevator, and probably some handles to hold onto. In theory, things like life support and sleeping accommodation could be housed in a mobile elevator car enclosing the cyclist, so that while s/he pedals upwards, the car moves upwards to follow.
Finally we're ignoring the fact that a human being standing at the equator has some kinetic energy due to the rotation of the Earth, and that a human being standing on a geostationary platform 100km above the equator has a little more kinetic energy (same angular speed, longer circumference). I ran the numbers and it only adds about 20 minutes of pedalling time.