People seem very confused here. The original question is whether it makes a difference buying a stock via an ADR (and so paying for it in dollars) or buying it on its native exchange (and so paying for it in its native currency). The answer to that is that it basically makes no difference at all (modulo some possible issues with bid/ask spreads). Arbitrage keeps the dollar value of an ADR pretty much the same as the value of one (native) share when converted into dollars.
The more complex issue is what happens to a foreign stock when the exchange rate shifts against the dollar. If the company is primarily an exporter (with dollar-linked prices), then likely you would do well if the native currency weakens against the dollar. There are two countervailing effects - you as a foreign holder effectively will show an immediate f/x loss, but on the other hand the company's earnings will likely benefit considerably, so this is likely to be more than offset by an increased stock price.
A concrete example might be Ono Pharmaceuticals. Their forthcoming PD-1 royalties are presumably dollar-denominated, and so any Yen weakness would benefit their earnings. They have domestic sales as well, so it might end up as something of a wash.
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