Illescas rapidly became Spain's strongest and most consistent player over many years, registering his country's highest ever Elo rating (2620) in 1993, making him at the time, world number 26.

As a young man, Illescas' tournament results were noteworthy; 1st= Las Palmas 1987 and 1988, 1st Oviedo 1991, 1st= Pamplona 1991/92 (with Leonid Yudasin), 2nd= Leon 1992 (after Boris Gulko), 3rd Chalkidiki 1992 (after Vladimir Kramnik and Joël Lautier), 1st Lisbon Zonal 1993 and 2nd Wijk aan Zee 1993 (after Anatoly Karpov).

In 1997, he was appointed to the IBM-led team that prepared the super-computer Deep Blue in the build up to a second match with Garry Kasparov.

Working with Joel Benjamin, Nick DeFirmian and John Fedorowicz, the project and match result were an unreserved success and this undoubtedly enhanced his reputation as an analyst, team player and perhaps most importantly, someone who understood the psyche of the incumbent world champion better than most.

With the arrival of super-grandmaster Alexei Shirov in 1994 and the subsequent emergence of Vallejo Pons as a world-class grandmaster, Illescas may no longer be Spanish number one, but he remains a tough and widely respected competitor.