Originally posted by Nasedkina Question: if this kind of tourist wants to stay longer then 90 days, will the Federal Police grand him additional 90 days?

Usually YES, you just have to show a returno or ongoing ticket, proofing you´ll leave Brazil.

Originally posted by Nasedkina Or, does this tourist need to make a quick "visa run" to leave Brazil and come back for additional 90 days? Will he/she be allowed to enter Brazil again, after his first 90 days are over?

That´s another option, but I guess only interesting for thoose living or staying close to the border, or thoose who planed already to make a quick visit to Argentina, Paraguay or so.

Originally posted by Nasedkina In general, are additional 90 days in a row possible for a tourist who doesn't have/need a special tourist visa to be in Brazil?

First, the visa extention applies to ALL foreigners, no matter if you had to get a visa through a Brazilian consulate BEFORE entering the country, or visa free.

But keep in mind that it´s possible that on your first entry the imigration won´t grant your the full 90 days !!! 90 days is the maximum, but it happens that they give you less days, so you need to check what the write down into the stamp in your passport.
Because it´s important to apply for the visa extention BEFORE it expires. They don´t give you additional days if the visa already expired.

Many tourists don´t pay attention to the granted time frame they are allowed to stay, because they think they will get the maximum of 90 days automatically.

It is not a visa extension. It is an extension of stay. So you don't necessarily need to have originally entered with a visa, like someone from the UK, who can enter with a visa exemption. But according to the chart sven points to not all EU citizens are so fortunate, and some, the French for example, are supposedly not eligible for the extension of stay. Check the list to see where you stand.
http://www.portalconsular.mre.gov.br/antes/quadro-geral-de-regime-de-vistos-1
An extension of stay is processed by the Policia Federal within Brazil and is for up to 90 days. It is at the discretion of the PF and there is a fee.
Visas are processed by Brazilian Consulates, outside of Brazil.

Best Luck!!!