Overhanging Stacks

After returning home from your latest expedition into the wilderness of Puzzlania, you seem to have brought back a surplus of circular reindeer bone tokens. One evening while sitting at your desk in the library, going over the notes for your book on the barbarians of the northern highlands, you idly play with the bone tokens as you think.
To amuse yourself, you start stacking the flat discs in various unorthodox ways. One stacking game consists of stacking the tokens in such a way that the topmost token is as far away from the bottom token as possible, when seen from above. That is, the horizontal distance between the top and bottom tokens should be as great as possible (the vertical distance is uninteresting).

One question strikes you as you stack and restack these tokens; what is the limit to how far you can offset the top token? Leaving your (frankly boring) notes for another evening, you pursue the solution to this puzzle.