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Buy 80 Endeavour space shuttles
NASA spent around $1.7 billion on Endeavour, the space shuttle orbiter built to replace Challenger. With $137.1 billion, Apple could just stretch it enough to buy 80 space shuttles.
But, of course, this doesn't include the costs of fuel, astronaut labor costs, and launching the thing into orbit. Realistically, Apple could probably buy around 50 at today's prices and still have plenty left over to run a half-decent space program of its own.
Plug less than 1 percent of the U.S.'s federal debt
The U.S. federal debt is a very contentious topic. Currently at $16.07 trillion, it's thought to be one of—if not the largest amount of money in the world. And, even with that, nobody has it. That amount of money doesn't actually exist, only in the vague subconscious minds of bankers.
Apple could help its good friend out—the U.S. government—by plugging everything it has into the U.S. federal debt. $137.1 billion into $16.07 trillion amounts to 0.8 percent. That's it. It's barely a drop in the ocean, but it's a start.
Take out its cash in dollar bills and stack it into space
Apple could reach space without a shuttle or a rocket—it just needs a very, very large ATM.
A one dollar bill has the dimensions of about 2.61 inches wide and 6.14 inches long, but only 0.0043 inches thick. Stack up $137.1 billion in dollar bills and that's 589.53 million inches, or 14,974 kilometers. (For imperial fans, that's 9,304.4 miles.)
The distance between Earth and the Moon varies from 221,000 miles to 252,000 miles—so the dollar bill pile won't quite reach the Moon, but it will pass the International Space Station's orbiting height of about 230 miles (370km) about forty-times over.
It's a moot point anyway; there aren't (and never has been, nor ever will be) 137.1 billion single dollar bills in circulation.