There's a few options but I'm not sure if there are any "great" ones. I take it Firebug/IE Dev toolbar profiling won't work because you are trying to track remote user's actions.
So, one option (I'm not highly recommending for production purposes), will work in some but not all browsers.
Essentially you overwrite every function, with a wrapper that you then inject your logging.
(I haven't tested this, trying to recall it from memory... hopefully in "pseudo code" you get the idea...)
//e.g. get all functions defined on the global window object
function logAll(){
var funcs = [];
var oldFunc;
for(var i in window){
try {
if(typeof(window[i]) == 'function'){
if(i != 'logAll'){
funcs.push(i);
}
}
} catch(ex){
//handle as desired
}
}
var x;
for(var i in funcs){
x = '_' + new Date().getTime();
window[x] = window[i];//save the old function as new function
//redefine original
window[i] = function(){
//do your logging here...
//then call the real function (and pass all params along)
call(window[x]);
};
}
};