This post compares coins from twelve different currencies. The data were pulled from Wikipedia and cross-checked with Numista.
Australia has the fattest single coin, both in absolute terms and also in proportion to its diameter. Its $2 coin is 3.2mm thick and only 20.5mm in diameter, making it less than 7 times long as it is thick. The average coin is ~12.8 times as long. For comparison, a US nickel (5 cents or $0.05) is both thinner (1.95mm) and wider (21.21mm).
The skinniest coin could be the South Korean 10 won, as it is only 1.2mm thick. Or it could be the Mexican 2 peso, because it is only 1.4mm thick while being 23mm wide - more than 16 times wider than thick.
The currency with the fattest coin on average is the Euro, with diameters only 10.8 times as much as thickness. South Korea has the thinnest coins, with an average proportion of 14.6 among coins of the won.
As a bonus, I found data for several currencies detailing how many of each coin is in current circulation. The data source is the Bank for International Settlements. Using this, we can see how close each currency gets to reaching the moon.
Unfortunately, they do not have data for the United States, although they list an estimate of 50.28 billion in total value. According to mintage figures,1 most US coins produced in the last 50 years are pennies (60%), much more than nickels (9%), dimes (15%) and quarters (15%). Halves (0.1%) and dollar coins (0.8%) are not often circulated.
This estimate strikes me as unreasonable. If looking at coins made since (say) 1977 and assuming a life expectancy of 30 years,2 then only about 23.89 billion in coin value would be in circulation.
The small estimate would put the US coin stack roughly on par with the Euro, while the larger estimate would send it past the moon.
See https://en.wikipedia.org/wiki/United_States_Mint_coin_production
Formula for surviving coins of a given denomination:
production × exp( (then-now) / 30 )