That really is what determines what price you pay.

If you simply sit above the current highest bid, eventually someone will hit you because someone will sell at market rate eventually (and that's your bid).  What could happen is the exchange rate rises though and you have to stay above the highest bid and you could end up paying more than if you hit the lowest ask right off the bat.

So it all depends what you mean by "relatively quickly".  Since you don't when a sell order at market will come in that could be hours or more.  Or it could be ten minutes.