In the very long term, we may replace the asteroid belt with thinsats for solar-system-scale computation. They will convert a significant fraction of sunlight into heat, effectively heating the night sky. How many can we add, and how much power is involved?
The Earth has a black body temperature of 254K, and an average "radiant exitance" of 237 W/m2 to the deep sky, which has an average temperature of 2.7K producing negligable back-heating. If the sky radiates heat back to the earth, the equilibrium earth temperature must increase to maintain heat balance. Assume that the earth can tolerate a 1K increase in black body temperature (to 255K), perhaps in conjunction with other technological mitigations to reduce environmental impacts. That would correspond to a radiant exitance of 240.75 W/m2, or an additional 3.75 W/m2 of heat from the asteroidal thinsat array.
Assume further that the thinsats are distributed throughout the sky, and intercept some fraction of total solar output. At asteroidal distances (2.7 A.U.), they will be intercepting about 200 W/m2, and 100W/m2 of the resulting heat from their 200K surfaces will be inbound towards the inner solar system, times the fraction of the sky covered by thinsats. The result is that 3.75% of the Sun's total output can be intercepted, or 3.75% of the sky can be covered by thinsats. This can be increased if the thinsats are designed for higher backside than frontside radiation, perhaps by the addition of perpendicular radiators.
The total solar output is 3.84E26 Watts, so the total light intercepted by the thinsats can be more than 14 trillion terawatts. This is 80 million times the solar energy input to the Earth.
Assuming advanced molecular self-assembly, tuned light-to-electrical efficiencies will be higher, and the lower temperature permits lower energy per bit. It is difficult to extrapolate how much computing is possible under these circumstances, but it may be trillions of times the total supportable near the earth.