How high can you count?
JavaScript (ES7), score 17, 176 bytes

They said it couldn't be done, so I did it :D (thanks to a lot of help from @FullDecent)

""**""
~(~/~//~/~/)/~/~/
3
4
!NaN- -!NaN- -!NaN- -!NaN- -!NaN
6
7
8
9
++[[]][+[]]+[+[]]
11
'lengthlength'.length
222&gt;&gt;2&gt;&gt;2
`${``^``}xE`^``
0XF
C=CSS==CSS;C&lt;&lt;C&lt;&lt;C&lt;&lt;C&lt;&lt;C
555555555555555555555%55


Unused characters:

#&amp;,:?@ABDGHIJKLMOPQRTUVWYZ\_bcdfijkmopqrsuvwyz|


I don't think 18 is possible, but I said the same thing about 17...

Explanation

JavaScript is a very weakly typed language; if you try to perform a mathematical operation on a non-number value, JS will try its hardest to convert it to a number first. This allows for a lot of interesting solutions. I've tried to avoid using the digits as much as possible so they can be used later on.


** is the exponentiation operator in ES7. The empty string when coerced to a number becomes 0, so this calculates 0 ** 0, which is 1 according to JavaScript.
A little ridiculous, but it works. /~/ is a regex literal, and ~/~/ returns -1, so this is ~(-1 / -1) / -1 = ~(1) / -1 = -2 / -1 = 2. (credits to @GOTO0 for the idea) 
Simply 3.
Now simply 4.
NaN is falsy, so !NaN is true, which is equivalent to 1. The expression thus becomes 1 - -1 - -1 - -1 - -1 = 1 + 1 + 1 + 1 + 1 = 5.
Simply 6.
Simply 7.
Simply 8.
Simply 9.
This uses a little JSF*** magic. +[] is 0, so [[]][+[]] returns the first element of [[]] (that is, []), and ++ increments this to 1. Then +[+[]] adds the array [0], which is coerced to a string and makes "10".
Simply 11. I originally had used 11&amp;1111 for 3 and 33/3 for 11 until I realized yet again just how dumb I am...
This strategy would work on any number: create a string of length 12 and use .length.
I just messed around with 2s and &gt;s for a while to get this one. I got lucky again: 222 &gt;&gt; 2 is 55, and 55 &gt;&gt; 2 is 13.
This one is rather tricky. The basic idea is to create 14 in hex (0xE), but we need the digit 0 elsewhere. So instead we prepend the result of &#96;&#96;^&#96;&#96; to the string xE; the empty string coerced to a number is 0, so this returns 0 ^ 0 = 0. Then the result is XORed with the empty string, which converts both values to numbers; "0xE" ^ "" is 14.
This is now pretty easy: 0XF is a hexadecimal literal with a value of 15.
The trickiest of all? First we set the variable C to CSS == CSS (that is, true). Then we take the result and perform &lt;&lt; C four times, which basically multiplies 1 by 2 ** 4.
JavaScript starts to lose integer precision at 253, which allows 555...555%55 to return a number that's not 0 or 5. I happened to get very lucky while playing around here.


Strategies that would work on many numbers:


-!NaN would work on any number (currently 5), though it gets large very fast.
~/~/ would work on any number (currently 2), though it gets very large very fast.
+[] would work on any number (currently 10), though it's by far easiest on 10 or 11.
.length would work on any number.
&#96;${-&#96;&#96;}xE&#96;-&#96;&#96; would work on pretty much any number if you do it right.
C&lt;&lt;C would work on any power of two (currently 16), or any int at all if you included |.