The Portuguese coach signed a new deal with the Primera Liga side that will keep him at the club until June 2016 after much speculation that he was looking to return to England.
"At this mature level of my career, I need big challenges which force me to still try to be the best," Mourinho said.
To remain with Real "seems to me to be the perfect challenge for the coming four years".
"We must improve as a team and also as individuals, and we must play a football that aims not only at winning matches but also winning over lovers of the game."
Mourinho added in typically brash style: "We have played the best football in the world even if we didn't win the Champions League.
"But we did win La Liga, which is more difficult, by playing a fantastic style of football, something which I think is the most important."
Mourinho hinted at a couple of off-season transfers for a squad he deemed remained "young".
"It's not a team on the way to extinction," he stressed. "It's not a team that's playing its last high-level footballing years, quite the opposite. It's a team that has everything to go for.
"With a couple of transfers to improve the squad we have, the team of 100 points (in the league), which I think will remain in the history of Real Madrid, deserves all our trust for next season."
The Champions League will be Mourinho's big target after his first two years in Madrid ended in two eliminations at the semi-final stages.
Real striker Cristiano Ronaldo praised Mourinho in an interview in Spanish sports daily AS, calling him "the best coach in the world".
"It's great to work with him because, right now, he is the number one manager," Ronaldo said.
Follow us @SprtNationalUAE