I bought a lamp that has a metal shade. The sticker says max 60 watt but it clicks 3 times like a 3-way with the light coming on on the second click. How do I determine what max 3 way bulb will be safe?
If it says 60 watts max, then the maximum wattage bulb you can safely use is 60 watts. If you could find a 3-way incandescent bulb like a 20-40-60, that would be safe. But I've never seen one that small.
With such a low acceptable wattage, the lamp may have been designed with CFL or LED lamps in mind, you can get something like an 11-23-34 Watt CFL 3-way bulb that gives light equivalent to a 50-100-150W incandescent. That would be perfectly safe in this lamp, assuming that it fits inside the "harp" or whatever holds the shade on.
3-way LED's are a little harder to find, but they do exist and they often are closer to a standard light bulb in size and shape, so they may fit better in your lamp.