Sure, get a little 8-pin Microchip microprocessor, like the 12F675, and program it to do just that. Program the number of times you want to turn your device on//off into the PIC's EEPROM. Everytime someone presses the button, the PIC turns the device on and subtracts one from that value. Press the button again to shut things off, without changing the number. When the value reaches zero, it could nothing or flash an LED or sound a warning buzzer to tell the user that all the uses are done. Now you must plug in a serial cable to a PC to reprogram a new value into the EEPROM again. Or enter a 4-digit PIN on a builtin keypad to reinstate the count again. Or use a KEYLOCK device to do this. Whatever does the job for you. This is the easiest way to go, parts-wise. But you need programming experience and a programmer.
At minimum, you could use an IC like the 14526 count-down timer. One of these will let you preset it to 15 uses. Press a button, subtract one. Cascade two and you can program them for up to 255 uses. Put a small connector on the board that allows you to reprogram it//them in the future. You'll have to add circuitry to shut off the device without affecting the count. This is all hardware, but can be trickier to implement in just logic and timing ICs.
Now determine how much power you are switching, and determine if isolation is necessary, then design in the appropriate on-off circuit.
And you're done!
kenjj