The documentation page: https://www.arduino.cc/en/Tutorial/Debounce has an error in the code.

The variable buttonState at line 40 is uninitialized which will cause undefined behaviour the first time line 75 if (reading != buttonState) is executed.

Usually (based on testing) buttonState is not equal to reading which results in detection of a button press on startup that did not happen. Since reading is HIGH when the button is not pressed, the correct initial value for buttonState is also HIGH. When changing this, the example functions correctly and the LED is on after the program started.