The biggest government expenses in the United States:
- Social Security.
If you require people to save for their own retirement, that takes away the two biggest expenses. If your country is not worried about defense, that's another.
Finance education with loans. So people go to school for however long and then pay back the loans once they start working. This will work much like a tax, but it's technically not a tax.
Finance law enforcement as mandatory insurance (it's also possible to fund healthcare that way). The insurers will handle the police duties to keep payouts down.
So now we're down to your initiatives. Caring for the truly indigent is not one of the four big expenses. Nor is art. Nor are giant machines. These might be funded by seigniorage. That's the amount of profit from printing money. Currently the US uses a fractional reserve system, where 90% of that value is lost. Your country might use a full reserve system, so the government captures the full value. This would cover about 10% of government expenses in the US, but your country has lower expenses. It might manage 40% or 50%.
The government may own all coin metal mining rights. So anyone who wants to buy platinum, gold, or silver (or copper?) must buy it from the government.
If that's still not enough, consider a program of mandatory service. So one day a week, each citizen has to work for the government. The citizen could help care for the indigent, make art, build giant machines, mine coin metal, or enforce regulations or laws. Or perhaps allow citizens to buy their way out. Again, that would work like a tax but would technically not be a tax.
Mandatory service could also replace unemployment and able-bodied welfare. Everyone works if they can. During recessions, the government would have more money, as they'd open more mines with the additional labor.
Charge products a fee based on the added costs they add. Food producers pay for anti-obesity programs (sugar and butter makers may pay more than oatmeal producers). Beer, wine, and liquor producers pay for alcoholism treatment and some medical care (e.g. cirrhosis treatment) that way. Cigarette sellers pay for programs to quit smoking and cancer care. Other drug purveyors pay for anti-addiction treatment. This fee would work much like sin taxes but again is technically not a tax.
All roads are toll roads or private roads. The tolls (not taxes) pay for those roads and the residents pay for private roads through homeowners' associations. Toll roads get you from place to place. Private roads get you to residences and in some cases, businesses.