Rodrygo, 18, agreed to join Madrid last year shortly after breaking into the first team at Santos, the Brazilian club that produced Pele and Neymar.
Last year he became the youngest player to score in the Copa Libertadores, South America's equivalent of Europe's Champions League, missing school to net for Santos against Uruguayan side Nacional.
Rodrygo joins Madrid a year after fellow Brazilian teenager Vinicius Jr moved to the Spanish capital, storming into the first team after spending his first few months with Real's reserve side Castilla in the third tier of Spanish football.
Madrid's latest arrival said he was prepared to do the same.
"Madrid will always have the best players in the world and I'm ready to do whatever the club wants, to play in the first team or Castilla," Rodrygo said at his presentation at Madrid's Santiago Bernabeu stadium.
"Here the pressure is bigger than anywhere else and Vinicius has told me that I have to be patient. But I'm young and I will be. Things are going to develop naturally."
SQUAD REBUILD
Rodrygo is the fifth player to join Madrid in the close season as part of an ambitious squad rebuild that has already cost more than 300 million euros.
Real are looking to make amends for a woeful last campaign in which they finished an unprecedented 19 points adrift of La Liga champions Barcelona.
Coach Zinedine Zidane has signed Brazilian defender Eder Militao for 50 million euros, Serbian striker Luka Jovic for 60 million, French right back Ferland Mendy for 48 million and Belgian forward Eden Hazard for 100 million.
Rodrygo said he did not expect his first season at Madrid to be plain sailing but was excited to be joining the 13-times European champions.
"I'm in my first season in Europe and there might be difficult moments but I'm prepared for that and I'm sure I'm going to have a lot of happy moments," he added.
"I always told my dad that one day I was going to play for Madrid, I told my agents this and the deal was completed in 20 minutes."
(Reporting by Richard Martin; Editing by Ken Ferris)
