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Regex to allow certain special characters - escape issue

I've working on a javascript regex that I intend to use with the jquery validate plugin (I'll add this as an additional method). It must (among other rules):
test if at least one of the following special characters is entered:
!, ", #, $, %, &, ', (, ), *, +,-, .,/, :, ;, <, =, >, ?, #, [, \, ], ^, _, `, {, |, }, ~
not allow 3 consecutive identical characters:
passed:
aa
99
++
not passed:
aaa
999
+++
The problem with my regex is that is having problem with these mentioned rules:
I think the issue is related to escaping and I've tried escaping + and - to no avail. Can anyone help! This is my regex: http://regexr.com/3ack3

This is one of those requirements where you can really simplify your life by using multiple regexes, rather than trying to cram all the logic into one complex regex with many assertions. Here's some JavaScript that implements your requirement:
var specialCharRegex = /[!"#$%&'()*+.\/:;<=>?#\[\\\]^_`{|}~-]/;
var threeConsecutiveRegex = /(.)\1\1/;
var input = prompt();
if (specialCharRegex.test(input) && !threeConsecutiveRegex.test(input)) {
alert('passed');
} else {
alert('failed');
} // end if
http://jsfiddle.net/t8609xv2/
Some notes on the trickier points:
inside the bracket expression, the following four special characters had to be backslash-escaped: /[\]. (Forward slash because it delimits the regex, backslash because it's the escape character, and the brackets because they delimit the bracket expression.)
inside the bracket expression, the dash had to be moved to the end, because otherwise it would likely specify a character range. When at the end, it never specifies a range, so it's always safer to put it there.
This modular approach also benefits maintainability, as you will more easily be able to make changes (modify/add/remove regexes, or change the if-test logic) at a later point in time.
Another benefit is that you could test each regex independently, which could allow you to provide a more accurate error message to the user, as opposed to just saying something like "invalid password".
Edit: Here's how you can whitelist the chars that are accepted in the input:
var specialCharRegex = /[!"#$%&'()*+.\/:;<=>?#\[\\\]^_`{|}~-]/;
var threeConsecutiveRegex = /(.)\1\1/;
var nonWhitelistCharRegex = /[^a-zA-Z0-9!"#$%&'()*+.\/:;<=>?#\[\\\]^_`{|}~-]/;
var input = prompt();
if (specialCharRegex.test(input) && !threeConsecutiveRegex.test(input) && !nonWhitelistCharRegex.test(input)) {
alert('passed');
} else {
alert('failed');
} // end if
http://jsfiddle.net/t8609xv2/2/

^(?=.*[!"#$%&'()*+,,\/:;<=>?#\[\]^_`{|}~-])(?!.*(.)\1\1).*$
Try this.See demo.
https://regex101.com/r/wX9fR1/10
You need a positive lookahead to check for special characters.
And
A negative lookahead to check if a character is is there 3 times.

You can use this regex:
^(?!.*?(.)\1{2})(?=[^a-z]*[a-z])(?=[^A-Z]*[A-Z])(?=\D*\d)(?=.*?[!##$%^&*()_=\[\]{};':"\\|,.<>\/?+-]).{8,20}$
RegEx Demo
You might be able to shorten it using:
^(?!.*?(.)\1{2})(?=[^a-z]*[a-z])(?=[^A-Z]*[A-Z])(?=\D*\d)(?=.*?[\W_]).{8,20}$
i.e. using non-word property \W instead of listing each and every special character.

Identifying special non alphanumeric characters in a string

Im working on a password validation that should only allow a-z 0-9 and these characters "!"#$%&'()*+,-./:;<=>?#[\]^_{|}~`
I tried using a regex but I'm not too good with them and I wasnt sure if this is even possible or if Im not escaping the correct characters.
var allowedCharacters = /^[A-Za-Z0-9!"#$%&'()*+,-.\/:;<=>?#[\\]^_`{|}~]+$/;
if (!s.value.match(allowedCharacters)){
displayIllegalTextError();
return false;
}

You need to place the dash at the start or end of the regex, or it will try to create a character range (,-.). Then, a-Z isn't a valid range, you probably meant a-z. Also, you need to escape the closing brackets:
/^[A-Za-z0-9!"#$%&'()*+,.\/:;<=>?#[\\\]^_`{|}~-]+$/

Looking over the ascii chart here I see your regex could be reduced to this character range:
/^[\x21-\x7e]+$/
If you just want to learn special behavior of character classes, you should read up
on it via regex basic tutorials.
Note that class behavior differs amongst the different flavors.

Simpler and more to the point using unicode: ^[\u0021-\u007E]+$.
/^[\u0021-\u007E]+$/.test('MyPassword!') // returns true
/^[\u0021-\u007E]+$/.test('MyPassword™') // returns false
Now if you would like to go a few steps further and actually create a more complex validation such as: minimum length 8 characters and at least one lowercase, one uppercase, one digit and one special character:
^(?=.*[a-z])(?=.*[A-Z])(?=.*[0-9])(?=.*[^a-zA-Z0-9])[\u0021-\u007E]{8,}$

regex to strictly check alphanumeric and special character

To check alphanumeric with special characters
var regex = /^[a-zA-Z0-9_$#.]{8,15}$/;
return regex.test(pass);
But, above regex returns true even I pass following combination
asghlkyudet
78346709tr
jkdg7683786
But, I want that, it must have alphanumeric and special character otherwise it must return false for any case. Ex:
fg56_fg$
Sghdfi#90

You can replace a-zA-Z0-9_ with \w, and using two anchored look-aheads - one for a special and one for a non-special, the briefest way to express it is:
/^(?=.*[_$#.])(?=.*[^_$#.])[\w$#.]{8,15}$/

Use look-ahead to check that the string has at least one alphanumeric character and at least one special character:
/^(?=.*[a-zA-Z0-9])(?=.*[_$#.])[a-zA-Z0-9_$#.]{8,15}$/
By the way, the set of special characters is too small. Even consider the set of ASCII characters, this is not even all the special characters.

The dollar sign is a reserved character for Regexes. You need to escape it.
var regex = /^[a-zA-Z0-9_/$#.]{8,15}$/;

What does /;/ and /^ +/ denote?

I recently came across the statement :
var cookies = document.cookie.split(/;/);
and
var pair = allCookies[i].split("=", 2);
if (pair[0].replace(/^ +/, "") == "lastvisit")
In the first statement what does /;/ in the argument of split denote ?
In the second statement what does /^ +/ in the argument of replace denote ?

These are Regular Expressions.
Javascript supports them natively.
In this particular example:
.split(/;/) uses ; as the split character;
.replace(/^ +/, "") removes ("") any (+) leading (^) whitespace ().
In both examples, / surround or delimit the regular expression (or "regex"), informing Javascript that you're providing a regex.
Follow the links provided above for more information; regexes are broad in scope and worth learning.

Slashes delimit a regular expression, just like quotes delimit a string.
/;/ matches a semi-colon. Specifically:
var cookies = document.cookie.split(/;/);
Means we split the document.cookie string into an array, splitting it where there are semicolons. So it would take something like "a;b;c" and turn it into ["a", "b", "c"].
pair[0].replace(/^ +/, "")
Just strips all leading whitespace. It turns
" lastvisit"
into
"lastvisit"
The caret ^ means "beginning of line", it's followed by space, and the + means to repeat the space one or more times, as many as possible.

The // syntax denotes a regular expression (also known as a 'regex').
Regex is a syntax for searching and replacing strings.
The first example you gave is /;/. This is a very simply regex which just searches the string for semi-colons, and then splits it into an array based on the result. Since this is not using any special regex functionality, it could just as easily have been expressed as a simple string, ie split(";") (as has been done with the equal sign in your other example), without making any difference to the result.
The second example is /^ +/. This is more complex and requires a bit of knowledge of how regex works. In short, what it is doing is searching for leading spaces on a string, and removing them.
To learn more about regex, I recommend this site as a good starting point: http://www.regular-expressions.info/
Hope that helps.

I think that /^ +/ means: one or more no-" " characters

Javascript String pattern Validation

I have a string and I want to validate that string so that it must not contain certain characters like '/' '\' '&' ';' etc... How can I validate all that at once?

You can solve this with regular expressions!
mystring = "hello"
yourstring = "bad & string"
validRegEx = /^[^\\\/&]*$/
alert(mystring.match(validRegEx))
alert(yourstring.match(validRegEx))
matching against the regex returns the string if it is ok, or null if its invalid!
Explanation:
JavaScript RegEx Literals are delimited like strings, but with slashes (/'s) instead of quotes ("'s).
The first and last characters of the validRegEx cause it to match against the whole string, instead of just part, the carat anchors it to the beginning, and the dollar sign to the end.
The part between the brackets ([ and ]) are a character class, which matches any character so long as it's in the class. The first character inside that, a carat, means that the class is negated, to match the characters not mentioned in the character class. If it had been omited, the class would match the characters it specifies.
The next two sequences, \\ and \/ are backslash escaped because the backslash by itself would be an escape sequence for something else, and the forward slash would confuse the parser into thinking that it had reached the end of the regex, (exactly similar to escaping quotes in strings).
The ampersand (&) has no special meaning and is unescaped.
The remaining character, the kleene star, (*) means that whatever preceeded it should be matched zero or more times, so that the character class will eat as many characters that are not forward or backward slashes or ampersands, including none if it cant find any. If you wanted to make sure the matched string was non-empty, you can replace it with a plus (+).

I would use regular expressions.
See this guide from Mozillla.org. This article does also give a good introduction to regular expressions in JavaScript.

Here is a good article on Javascript validation. Remember you will need to validate on the server side too. Javascript validation can easily be circumvented, so it should never be used for security reasons such as preventing SQL Injection or XSS attacks.

You could learn regular expressions, or (probably simpler if you only check for one character at a time) you could have a list of characters and then some kind of sanitize function to remove each one from the string.
var myString = "An /invalid &string;";
var charList = ['/', '\\', '&', ';']; // etc...
function sanitize(input, list) {
for (char in list) {
input = input.replace(char, '');
}
return input
}
So then:
sanitize(myString, charList) // returns "An invalid string"

You can use the test method, with regular expressions:
function validString(input){
return !(/[\\/&;]/.test(input));
}
validString('test;') //false

You can use regex. For example if your string matches:
[\\/&;]+
then it is not valid. Look at:
http://www.regular-expressions.info/javascriptexample.html

You could probably use a regular expression.

As the others have answered you can solve this with regexp but remember to also check the value server-side. There is no guarantee that the user has JavaScript activated. Never trust user input!