The short answer, if using copper, is 0 AWG, commonly referred to as 1/0 or "one ought".
Type of wire depends on location & temperature, but in my common scenario I am using 1/0 THHN inside a 2" schedule 40 PVC conduit (3 - 1/0 wires plus a 4 ga bare ground) to run indoors, a distance of about 25 feet from my main panel to a sub for upstairs.
I could go into more detail regarding outdoor, direct burial, grounding, aluminum wire, etc but would need more details.
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A #2 AWG copper conductor with a insulation factor of 90 degrees C is rated at 130 amps.
A #3 copper conductor with an insulation factor of 90 degrees C is rated at 130 amps.
A 100 amp residential service requires a size #8 copper wire, it should be insulated in green.
2 copper
A 200 amp service panel with a 60 amp sub-panel.
For Service Entrance Cable 600 kcmill (Thousands of circular mils).
Yes, the wire size has to be increased. The existing 100 amp service will now probably have a #3 conductor with an insulation factor of 75 or 90 degrees C which is rated for 100 and 105 amps respectively. Increasing to a 150 amp service will use a #1/0 wire size. If you are thinking of an upgrade, skip the 150 amp service and go to a 200 amp panel. New homes are installing 200 amp 42 circuit panels. The extra cost is only reflected in the materials as the labour costs will be the same regardless whether a 150 amp or a 200 amp panel is installed. The wire size for a 200 amp panel is #3/0. With an insulation factor of 75 or 90 degrees C the rating is 200 or 210 amps respectively.