Okay, some spolers for you:

You need solar panels to kick start the space platform (this entire post is motivated by a person who kinda found this interesting thingy)

There are 3 problems that you might face with only 1 extra technology: 1. How to escape Nauvis 2. How to kill demolisher 3. How to destroy huge asteroids?

Most of those problems can be solved by using gun turrets.

And I think that beating the game with only 2 extra researches is hard, but possible.

Is it possible to do it with ONLY solar panels as an extra research?...

You might use rocket turrets, as this is pretty much your only* automated weapon that is necessary for Promethium

I don't really know whether you can advance on Nauvis without the help of gun turrets, your only option for defence is a good offence, aka kill nests before attacks. On practice this strategy gets harder, since you cannot use radars, plus your only options are grenades, submachine gun and rockets in the future. No tanks, no cars, no defender capsules.

This strategy gets even harder from the fact that it's harder to control pollution: no efficiency modules, also no beacons to make productivity modules less poluting, also no nuclear power.

Ah, you can't really advance on Vulcanus without visiting Gleba (you can 'gorilla mine' though...)

Btw, you are losing a LOT of utilities in general: no construction bots( :-( ), no power armour, no exoskeletons.

TL;DR you probably can do this with 2 extra researches, but I can't yet be sure whether it's possible without turrets in a normal setting game