Add a "hook" to all AJAX requests on a page

NOTE: The accepted answer does not yield the actual response because it is getting called too early.

You can do this which will generically intercept any AJAX globally and not screw up any callbacks etc. that maybe have been assigned by any third party AJAX libraries.

(function() {
    var origOpen = XMLHttpRequest.prototype.open;
    XMLHttpRequest.prototype.open = function() {
        console.log('request started!');
        this.addEventListener('load', function() {
            console.log('request completed!');
            console.log(this.readyState); //will always be 4 (ajax is completed successfully)
            console.log(this.responseText); //whatever the response was
        });
        origOpen.apply(this, arguments);
    };
})();

Some more docs of what you can do here with the addEventListener API here:

https://developer.mozilla.org/en-US/docs/Web/API/XMLHttpRequest/Using_XMLHttpRequest#Monitoring_progress

(Note this doesn't work <= IE8)

Since you mention jquery, I know jquery offers a .ajaxSetup() method that sets global ajax options that include the event triggers like success, error, and beforeSend - which is what sounds like what you are looking for.

$.ajaxSetup({
    beforeSend: function() {
        //do stuff before request fires
    }
});

of course you would need to verify jQuery availability on any page you attempt to use this solution on.

Inspired by aviv's answer, I did a little investigating and this is what I came up with.
I'm not sure that it's all that useful as per the comments in the script and of course will only work for browsers using a native XMLHttpRequest object.
I think it will work if javascript libraries are in use as they will use the native object if possible.

function addXMLRequestCallback(callback){
    var oldSend, i;
    if( XMLHttpRequest.callbacks ) {
        // we've already overridden send() so just add the callback
        XMLHttpRequest.callbacks.push( callback );
    } else {
        // create a callback queue
        XMLHttpRequest.callbacks = [callback];
        // store the native send()
        oldSend = XMLHttpRequest.prototype.send;
        // override the native send()
        XMLHttpRequest.prototype.send = function(){
            // process the callback queue
            // the xhr instance is passed into each callback but seems pretty useless
            // you can't tell what its destination is or call abort() without an error
            // so only really good for logging that a request has happened
            // I could be wrong, I hope so...
            // EDIT: I suppose you could override the onreadystatechange handler though
            for( i = 0; i < XMLHttpRequest.callbacks.length; i++ ) {
                XMLHttpRequest.callbacks[i]( this );
            }
            // call the native send()
            oldSend.apply(this, arguments);
        }
    }
}

// e.g.
addXMLRequestCallback( function( xhr ) {
    console.log( xhr.responseText ); // (an empty string)
});
addXMLRequestCallback( function( xhr ) {
    console.dir( xhr ); // have a look if there is anything useful here
});