Double- and tripple clicking with buttom

Please use code tags!

This code part:

// only toggle the LED if the new button state is HIGH
      if (buttonState == HIGH) {
        ledState = !ledState;
      }

currently toggles the LED. Now change that to not toggle the LED but simple store the time:

// only toggle the LED if the new button state is HIGH
      if (buttonState == HIGH) {
        lastButtonPush = millis();
        pushCount++;
      }

and later on, just above the "set the LED" comment:

    if (pushCount && millis() - lastButtonPush > doubleClickTimeout) {
      if (pushCount == 2) {
        ledState = HIGH;
      } else if (pushCount == 3) {
        ledState = LOW;
      }
      pushCount = 0;
    }

In the global variables part you need to include this code:

uint8_t pushCount = 0;
uint32_t lastButtonPush = 0;
uint32_t doubleClickTimeout = 200; // or whatever you find appropriate