This answer mainly addresses the last part of the question "For Earth-based orbits, it appears that the farther away the satellite, the larger it is. I am just wondering if this will always be the case." (NB at the time of answering the question title was "Is it possible for small satellites to orbit in MEO or GEO" which has the simple answer of yes.)
The simple answer is it is likely but not inevitable that satellites launched to MEO and GEO may be larger.
The starting premise is that higher orbits require more energy to get to from the Earth's surface. Thereafter, talking in typical terms:
For delivery direct by the launch vehicle, this will cost more per
kg than for a LEO delivery, or the small satellite will have to find
a "ride-share opportunity". Small satellites are often small
because they are budget-constrained, or at least this has often been
true in the past, so the ride-share approach has been the only real
option for small satellites. There is no reason in principle why a 1kg cubesat could not have a direct rideshare delivery to MEO, GTO or GEO.
For delivery to an intermediate point the small satellite will need
its own propulsion system. For a small satellite to have some
potentially significant fraction of the mass is taken by propellant,
propulsion hardware, or solar arrays means that the satellite will
have less room for the primary payload.
Presently, plausible targets direct deliveries as rideshares are most likely to be GTO because of the large number of commercial missions there. MEO seems plausible with the traffic to deliver navigation satellites and there are even a small number of direct deliveries to GEO, usually on US and Russian government missions.
Examples of missions based on direct or near-direct rideshares:
The STRV 1a and 1b satellites (~50kg each) were deliberately launched on a rideshare into GTO as the target orbit so that they could measure the radiation environment and test experimental equipments. See here for further details.
The Spirale satellite (120 kg) was also launched into GTO on a rideshare but manoeuvered slightly to increase the perigee from 200km to 600km to prolong the mission life. The mission was an Early Warning demonstrator and its not entirely clear why it was based in near GTO, it seems quite possible that it was intended to be the forerunner of a GEO mission and used the apogee passes to obtain a GEO like view of the Earth. See here for further details.
Example: launch to GTO then manoeuvre to GEO
As a rough guide, a mission launched to GTO but targeted to manoeuvre to GEO under the power of the satellite could expect:
- half of its mass to be taken up by propellant if it used a high performance, e.g. 500N, chemical thruster. Such a thruster would probably have a mass of 5kg itself, a 50 - 100 litre propellant tank would be the same again.
- if a smaller chemical thruster were used, say 10N, its propellant efficiency and its manoeuvre efficiency would be worse still (scaling down of thrusters and longer thrust arcs respectively)
- if an electric propulsion system were to be used then the propellant mass could be very much reduced but instead a large solar array would be required - 2KW and upwards for the current commercial arcjets and plasma thrusters. Some scaling down is possible at the cost of longer transfer time.
It is entirely possible to do some rough calculations for the manouevres and equipment mass for such a mission or for other orbits though you can see there is a lot to consider; expect there to be a bit of a learning curve.
Short of actually stepping through such calculations its hard to see how this could add up looking like a large payload mass fraction for even a 50 - 100kg satellite, let alone a nanosat / cubesat.