I live in Germany, Hamburg
Beer is generally taxed in Germany.
Everyone is allowed to brew 200 L of tax free beer at home for private use.
Every year before or while the first batch is brewed you have to annouce that to the local customs office. Its easy, as you get an homebrewer number, and you can announce via email. You tell them you#re not aiming to brew more than 200 L this year. A few days later you get a confirmation per letter.
For any additional amount of beer you can fill out a customs form, calculate the taxes and send these small amounts via bank transfer to the customs office.
What country do you live in? Germany
Is beer as a product in general (commercial/non commercial) taxed in your country? Yes
Are you allowed to produce homebrew beer, that is tax free? Yes.
Is there some amount limitation of homebrew beer that you are allowed to produce without paying taxes? 200 liters.
Is there any legal obligation considering for instance an obligation to fill a report or announcement as a homebrewer and send it to your legal authorities in order to be allowed to brew beer at home?
According to the law, you need to report all brews to the authorities. Since homebrewing in Germany became a lot more common, most authorities (which are organized on the state level) only require you to report once per year that you intend to brew not more than 200 liters in your household.
Generally speaking: beer is taxed with EUR 0.787 per °P per hectoliter (100 liter = 1 hectoliter). Only full °P are counted. There exists a tax reduction for small producers of beer.
For homebrewers, the first 2 hectoliters are tax-free. Beyond that, there are reduced tax rates for both homebrewers and commercial brewers: for 5000 hectoliters or less, you only need to pay 56 % of the tax, and it goes up until 40000 hectoliters per year where the regular tax rate starts.
Example: imagine you already brewed your annual allowance of 200 liters of tax-free beer. You then brew a 20 liter batch of beer of 12.5 °P (1.050). So the tax that you'd need to report and pay is like this:
12 * 20 * 0.787 / 100 = EUR 1.88 is what you pay (times 12 because you only pay for full °P; lots of breweries abuse this and brew 11.9 °P beer to only pay taxes on 11 °P).
But because you've brewed less than 5000 hectoliters per year, you're eligible for the reduced rate of 56 %, so you really pay 56 * (12 * 20 * 0.787 / 100) / 100 = EUR 1.05 in taxes for your beer.
Here's the official explanation on the website of the tax authorities: