How long is my number?

Taxi, 1118 bytes

1 is waiting at Starchild Numerology.Go to Post Office:w 1 l 1 r 1 l.Pickup a passenger going to Chop Suey.Go to Chop Suey:n 1 r 1 l 4 r 1 l.Pickup a passenger going to Crime Lab.'-' is waiting at Writer's Depot.Go to Writer's Depot:n 1 l 3 l.Pickup a passenger going to Crime Lab.Go to Crime Lab:n 1 r 2 r 2 l.Switch to plan "n" if no one is waiting.-1 is waiting at Starchild Numerology.[n]0 is waiting at Starchild Numerology.Go to Starchild Numerology:s 1 r 1 l 1 l 2 l.Pickup a passenger going to Cyclone.Pickup a passenger going to Addition Alley.Go to Cyclone:e 1 l 2 r.[r]Pickup a passenger going to Cyclone.Pickup a passenger going to Addition Alley.Go to Zoom Zoom:n.Go to Addition Alley:w 1 l 1 r.Pickup a passenger going to Addition Alley.Go to Chop Suey:n 1 r 2 r.Switch to plan "f" if no one is waiting.Pickup a passenger going to Sunny Skies Park.Go to Sunny Skies Park:n 1 l 3 l 1 l.Go to Cyclone:n 1 l.Switch to plan "r".[f]Go to Addition Alley:n 1 l 2 l.Pickup a passenger going to The Babelfishery.Go to The Babelfishery:n 1 r 1 r.Pickup a passenger going to Post Office.Go to Post Office:n 1 l 1 r.

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Ungolfed:

1 is waiting at Starchild Numerology.
Go to Post Office: west 1st left 1st right 1st left.
Pickup a passenger going to Chop Suey.
Go to Chop Suey: north 1st right 1st left 4th right 1st left.
Pickup a passenger going to Crime Lab.
'-' is waiting at Writer's Depot.
Go to Writer's Depot: north 1st left 3rd left.
Pickup a passenger going to Crime Lab.
Go to Crime Lab: north 1st right 2nd right 2nd left.
Switch to plan "n" if no one is waiting.
-1 is waiting at Starchild Numerology.
[n]
0 is waiting at Starchild Numerology.
Go to Starchild Numerology: south 1st right 1st left 1st left 2nd left.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Addition Alley.
Go to Cyclone: east 1st left 2nd right.
[r]
Pickup a passenger going to Cyclone.
Pickup a passenger going to Addition Alley.
Go to Zoom Zoom: north.
Go to Addition Alley: west 1st left 1st right.
Pickup a passenger going to Addition Alley.
Go to Chop Suey: north 1st right 2nd right.
Switch to plan "f" if no one is waiting.
Pickup a passenger going to Sunny Skies Park.
Go to Sunny Skies Park: north 1st left 3rd left 1st left.
Go to Cyclone: north 1st left.
Switch to plan "r".
[f]
Go to Addition Alley: north 1st left 2nd left.
Pickup a passenger going to The Babelfishery.
Go to The Babelfishery: north 1st right 1st right.
Pickup a passenger going to Post Office.
Go to Post Office: north 1st left 1st right.

Explanation:

Pickup the input and split it into individual characters
Pickup the value 1.
If the first character a hyphen, add -1. Otherwise, add 0.
Keep picking up characters and adding 1 until you're out.
Convert the running total to a string and print to stdout.

Mathematica, 13 bytes

IntegerLength

There's a built-in... returns 0 for 0.

dc, 3

?Zp

Note that normally dc requires negative numbers to be given with _ instead of the more usual -. However, in this case, either may be used. If - is given, then dc treats this as a subtraction on an empty stack, throws dc: stack empty, and then continues with the rest of the number; Thus the result is no different.

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?    # input
 Z   # measure length
  p  # print