Edit There appears to be a better LZW solution that handles Unicode strings correctly at http://pieroxy.net/blog/pages/lz-string/index.html (Thanks to pieroxy in the comments).
I don't know of any gzip implementations, but the jsolait library (the site seems to have gone away) has functions for LZW compression/decompression. The code is covered under the LGPL.
// LZW-compress a string
function lzw_encode(s) {
var dict = {};
var data = (s + "").split("");
var out = [];
var currChar;
var phrase = data[0];
var code = 256;
for (var i=1; i<data.length; i++) {
currChar=data[i];
if (dict[phrase + currChar] != null) {
phrase += currChar;
}
else {
out.push(phrase.length > 1 ? dict[phrase] : phrase.charCodeAt(0));
dict[phrase + currChar] = code;
code++;
phrase=currChar;
}
}
out.push(phrase.length > 1 ? dict[phrase] : phrase.charCodeAt(0));
for (var i=0; i<out.length; i++) {
out[i] = String.fromCharCode(out[i]);
}
return out.join("");
}
// Decompress an LZW-encoded string
function lzw_decode(s) {
var dict = {};
var data = (s + "").split("");
var currChar = data[0];
var oldPhrase = currChar;
var out = [currChar];
var code = 256;
var phrase;
for (var i=1; i<data.length; i++) {
var currCode = data[i].charCodeAt(0);
if (currCode < 256) {
phrase = data[i];
}
else {
phrase = dict[currCode] ? dict[currCode] : (oldPhrase + currChar);
}
out.push(phrase);
currChar = phrase.charAt(0);
dict[code] = oldPhrase + currChar;
code++;
oldPhrase = phrase;
}
return out.join("");
}