Start with Tsiolkovsky equation. Pick a fuel - that determines the speed of exhausting gases. Choose the number of stages - since you're trying to reduce complexity, you can't rely on good mass ratios. Obviously those things are related.
When you'll try to make payload - and rocket - small, you'll find that some parts of the rocket reduce their mass in proportion to the total rocket mass, and some other parts don't. Tanks are usually the biggest parts by mass in an orbital rocket - and they scale down rather well, to a certain limit. Engines, however, could scale worse. That will affect your mass ratio, and at some point your payload will become zero.
This is the main problem with current state of the art in high power rocketry (if you want to launch a satellite, you have to have more than 8 km/s delta-V, and this is enough to qualify it as a high power rocketry). Some parts - like control circuits and some sensors - advanced greatly since late 1950-s, and you can have a whole control system in a really small and a lightweight package. But engines, valves - you'll have to have valves - and, after some point, tanks (because you can't have too thin of a wall without breaking in a real flight) will stop scaling. You'll likely find that it's about the same mass of a rocket which can launch 1/4 kg to orbit or 1 kg to orbit.
Elon Mask proved that $90 millions are enough to build a half-a-ton payload rocket from scratch. May be - may be - $1 million is enough to build and launch a smallest rocket from scratch. Hard to say... but it's definitely more than $100 thousands today.