Edit in JSFiddle

(function(){
    
    jQuery.fn.ajax = function(settings){
        
        var jQThis = this;
        
        function jQAjaxChain(settings){
            if(!(this instanceof arguments.callee)) {
                return new arguments.callee(settings);
            }
            
            var self = this, combined;
            
            this.callbacks = {
                 success: function(){}
                ,error: function(){}
                ,complete: function(){}
            };
            
            combined = $.extend({
                beforeSend: function(xhr, settings){
                    self.settings = settings;
                }
                ,success: function(){ self.callbacks.success.apply(self.settings, arguments); }
                ,error: function(){ self.callbacks.error.apply(self.settings, arguments); }
                ,complete: function(){ self.callbacks.complete.apply(self.settings, arguments); }
            }, settings);
            
            this.xhr = $.ajax(combined);
        }
        
        jQAjaxChain.prototype.success = function(callback){
            this.callbacks.success = callback;
            return this;
        }
        
        jQAjaxChain.prototype.error = function(callback){
            this.callbacks.error = callback;
            return this;
        }
        
        jQAjaxChain.prototype.complete = function(callback){
            this.callbacks.complete = callback;
            return this;
        }
        
        jQAjaxChain.prototype.exit = function(){
            return jQThis;    
        }
        
        return new jQAjaxChain(settings);
    }  
        
})();

var test = $('ul li')
    .ajax({url: '/echo/json'})
    .error(function(){ console.log('error', arguments, this); })
    .success(function(){ console.log('success', arguments, this); })
    .complete(function(){ console.log('complete', arguments, this); })
    .exit()
    .each(function(i){ console.log('element ' + i, this); });

console.log(test);

<ul>
    <li>Item 1</li>
    <li>Item 2</li>
    <li>Item 3</li>
</ul>