Solved by nils

solution: coin on a table

Here's what I came up with:

First, put the first quarter exactly in the center of the (perfectly circular) table.

Next, for each quarter the opponent places, place one directly opposite it. That is, place it so that the center of the table is halfway between your piece and the opponent's previous piece.

This will generate a completely symettric (about the center) layout of quarters on the table. This means that whenever the opponent selects a free space to place a quarter in, the space opposite is guaranteed to be free as well. Since we are always guaranteed an open space, we will never lose with this strategy (and thus win when there are finally no more spaces for the opponent to use).