I\'ve been trying to implement snook.ca\'s Simplest jQuery Slideshow, but when applied to child elements inside a instead of a straightforward stack
Just break your chaining after selecting the next li. If there is no next li, the length of the result set will be 0. When that happens, loop back to the beginning. Then, just finish your setup. Here is a revision on your JSBin code that shows it working:
$('header nav li').not('.current').children('p').hide();
setInterval(function(){
var $next = $('header nav li.current').children('p').hide()
.parent('li').removeClass()
.next('li');
if(!$next.length) $next = $('header nav li:first');
$next.addClass('current')
.children('p').show();
},3000);