score:5
From the US Customs and Border Protection website:
The United States regulation does NOT require a Visa Waiver Program (VWP) traveler to present a printed copy of the ESTA authorization page in order to travel. However, it is recommended that you print a copy for your records.
I carted around a printed copy for my first few visits to the US but it has never been useful and I have long since stopped.
That page does continue:
Some airlines may ask you to bring a copy, in which case you should print it out.
I think the key word here is ask - I never have been. Worst case, you could retrieve the details from the ESTA website even if you can't remember the application number, assuming you're travelling with a smartphone.