Anyway, while trying to get all of them I wondered just how long it'd take...
It's obvious that it'll take ages to get them all if you only spend 1 shell a time (about 730 attempts, assuming I've not gone horribly wrong somewhere), though it's surprising to see that that's actually the cheapest way of collecting them all. Playing to always win is the fastest with only 130 attempts needed, but chews through over six thousand shells. The game does throw vast quantities of shells at you, so that's possibly not as daft a method as it seems.
Excel gets very close with the equation for the Attempts needed line (it's actually y=-130ln(x)+130), but I'm unsure about the other one. It feels like it should also be logarithmic or exponential, but the only trend line that comes near it is a linear, or possibly squared function.