Exponential search is useful when you're querying a REST API that addresses resources with sequential IDs, and need the last ID, but there's no dedicated endpoint for it:

  HEAD /users/1 -> 200 OK
  HEAD /users/2 -> 200 OK
  HEAD /users/4 -> 200 OK
  ...
  HEAD /users/2048 -> 200 OK
  HEAD /users/4096 -> 404 Not Found

And then a binary search between 2048 and 4096 to find the most recent user (and incidentally, the number of users). Great info to have if you're researching competing SaaS companies.