Though Nero probably didn't fiddle while Rome burned (though Roman historian Suetonius reports that he "sang the whole of the 'Sack of Illium' in his regular stage costume"), he did benefit from the great fire of A.D. 64 that destroyed more than half the city. The blaze cleared an entire swath of Rome in the area where Nero had been constructing a relatively modest (for Nero) palace between two of Rome's seven hills.
After the fire, Nero began work on a reconfigured palace complex, this one to cover part to all of three hills of Rome: the Esquiline, Caelian, and Palatine, as well as the great valley in the middle. This new palace and grounds covered 81 hectares, or over 200 acres, in central Rome—the area now occupied by everything from the edge of the Circus Maximus to the Santa Maria Maggiore area, including the valley where the Colosseum sits.
The architecture of the buildings was revolutionary—for the first time, concrete was used with brio, creating massive barrel vaulting, an octagonal room, skylights, and supposedly a banquet hall with a revolving ceiling decorated to match the stars as it followed their progress across the skies. According to Suetonius, "In the rest of the house all parts were overlaid with gold and adorned with gems and mother-of-pearl. There were dining rooms with fretted ceils of ivory, whose panels could turn and shower down flowers and were fitted with pipes for sprinkling the guests with perfumes." The complex was named for its golden walls, Domus Aurea, or Golden House. In the vestibule stood a colossal statue, over a hundred feet (30 meters) tall, of Nero himself.
Nero's excesses did not endear him to the Romans, and this new project topped them all. After his death in 68, the house was abandoned and destroyed, and the colossal statue's head was changed to reflect the emperor of the era. During his reign, Emperor Hadrian moved the statue to the Flavian Amphitheater, which had been built on the drained land where Nero's lake had once stood. That immense stadium eventually became known as the Colosseum, not for its size, but for Nero's colossal statue.
For detailed information on the construction of the Domus Aurea, see Larry F. Ball's monograph, The Domus Aurea and the Roman Architectural Revolution, published by Cambridge University Press, 2003.