How many 2x4 Lego bricks would it take to build a real skyscraper?
Great question. Others may have better answers for a more "realistic" design, but the math for a solid structure is actually pretty quick. Take the Empire State Building for example:

Height: 381m
Base: 129m x 57m
Volume: 381m × 129m × 57m = 2 801 493m³


And a basic 2x4 brick:

Volume: 31.8mm × 15.8mm × 9.6mm = 4.823424e-6 m³


The number of bricks needed to simply fill the volume is total volume over brick volume, so:

Number of bricks: 2 801 493 / 4.823424e-6 = 580 810 022 092 bricks
Weight: 580 810 022 092 bricks * 2.32g = 1 347 479 metric tons


Basically, you're looking at 580 billion bricks weighing in at ~1.3 million metric tons for a solid structure the size of the Empire State Building.

This puts you at about a fifth of an Empire State Building's worth of bricks dumped in the ocean in 2014.