what is the maximum cable length to connect a tactile button to the Arduino?
What cable would be the best (awg)?
I dunno, but it's a looooooong way.
Do the math:
At 5V DC the AVR chip recognizes anything below 1V as "LOW" so you can afford to lose 4V over the wire.
The internal pullup resistors are about 30K Ohms so a pin will put out 0.15mA when configured as an input with pullup enabled.
Here's a table of wire gauges with resistances in ohms per kilometer (or kilofeet if you're still using old-fashioned units):https://en.wikipedia.org/wiki/American_wire_gauge#Tables_of_AWG_wire_sizes
You've now got two of the variables in Ohm's law (current and resistance) so the rest is easy.
V = R * 0.00015
Any value of R that gives an answer less than 4V over your required distance is good enough (though you may want to limit it to 1V just to be on the safe side...)