Using a non 3-way bulb in a 3-way lamp

I'm assuming this is a 100 watt equivalent bulb, that is actually only running 18-20 watts. Using a 100 watt equivalent will be safe as they run much cooler than even an incandescent.

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.