Side effects of running an attiny13a directly on 9v
I could just refer you to this answer of mine which relates to overclocking, but also covers supply voltage. But reuse of the image makes it (arguably) worth adding this answer.  

Do also refer to the answer linked to above !!! :-) .

Manufacturers provide specification sheets which indicate voltage ranges which a processor will work within, and "absolute maximum" voltage ranges over which the processor may not work correctly but will not suffer permanent damage. Applying voltages outside the "absolute maximum" range means that you risk doing permanent damage to the device. 
TYhe image below summaries what you can expect from your processor under various conditions of clock speed and voltage. The actual values are for an ATMega128 and may need to be adjusted for your device, but, the principle is the same - if you don't want to play hard ball with Dirty Harry, stay inside the Safe Operating Area. Anything to the right of this may be fatally fatal. Any other non-grey areas have expectations as labelled.