What is the maximum range for a weapon? 3 times it's recorded range? 4? 5?

It came up

-Nearyn

From the Fine Rulebook, p 127:

Range: This number is used to determine how far the weapon can be accurately fired in metres. A weapon’s Short Range is half this number while its Long Range is double this number (for ranged bonuses and penalties see Chapter VII: Playing the Game, pages 198-199). Weapons cannot be fired at targets more than four times their range distance.

So there it is. 4 times its range value.

I wonder how many tiny little references like that are hidden in the Rulebook that I am missing…

There's a gunsight and a weapon upgrade that can both extend the range and range increments a bit in Hostile Acquisitions for Rogue Trader, but the respondant is totally right. Four times printed range is the maximum at which you can make a shot.