Seeing 770-555-5555 on screen has always been a huge pet peeve for me. It really kills the suspension of disbelief for me.
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But in this context it'd be the first digit of the area code, with no country code being used because the call is within the US. There are no area codes in the the North American Numbering Plan that start with a 1.
"The syntax rules for area codes do not permit the digits 0 and 1 in the leading position."
My guess would be it's to avoid ambiguity with the fact that 1 is also the country code. If I recall correctly, historically, dialing the 1 was necessary for any long distance call (even if not international).