My cursory research of LEGO tells me that the manufacturing of plastic LEGO bricks began in Denmark in 1947. If we were somehow able to obtain every single LEGO piece ever created since 1947 and make a pile of them, roughly how big would the pile be? Would it fill 10 Olympic-sized pools, maybe?
It's definitely hard to give any accurate numbers, but let's try some rough estimates.
On one hand, user Grover from the EuroBricks forum points out (emphasis mine):
I have around 200,000 bricks in my collection. [...] I wound up with a bunch of [boxes with] dimension: 33.78 x 37.78 x 40.00 cm (~45L). If I pack the bricks into these until they are full, it takes about 20 boxes to hold the collection [...]. No idea what the weight is on that, but if you consider 10,000 pcs/45L of volume, that might give you a rough estimate.
The company has made 400 billion LEGO elements or 62 bricks for every person on the planet.
I'm assuming short-scale billions (109), not long scale billions (1012). Billions are confusing.
Putting the numbers together we've got 1 800 000 000 litres (or 1.8×109 litres, or 1.8 million cubic meters).
Punching that figure into wolfram alpha provides a few amusing comparisons: that's 1.7 Empire State Buildings, or a cube with 122-meter-long edges.
But you're not interested in cubes. You ask:
If we [...] make a pile of them, roughly how big would the pile be?
Assuming that your pile is a cone with its radius equal to its height (i.e. it has a 45° slope), and doing the numbers, that would be a cone 120 meters high and 240 meters wide.
You also ask:
Would it fill 10 Olympic-sized pools, maybe?
A standard-sized Olympic swimming pool is (at least) 50×25×2 meters or 2.5 million litres. Doing the math throws out a rough estimate of (at most) 720 standard-sized Olympic swimming pools.
Again: these are very rough estimates based on the information linked above. It's possible that there's a more accurate estimate if more reliable numbers are used.
In particular, it should be possible to do a query to the bricklink database, and get the amount of parts for sale, per part; that should give an idea of how the different parts are distributed (i.e. "how many 1x1 tiles are there for each 2x4 brick?"). Normalize those numbers to bump the total to the "400×109" figure. As bricklink has data about the dimensions of each part, it should be possible to get a good estimate, maybe getting rid of very large parts along the way. Do not forget about the packing fraction, too.