$('select') - use document.getElementsByTagName, then loop over the returned list
.change(function() {…} - check out advanced event registration model for browser differences
$(this).val() - simply this.value; you should use this even in jQuery
$(this).parent().parent() - get the parentNode of the element (two times)
.find("input.hidden") - this is a bit harder. You could use .querySelector[All], but that does not work in legacy browsers. jQuery adds lots of sugar with its cross-browser selector engine. You might use another way to get the input element(s) that works cross-browser; you might try something along javascript document.getElementsByClassName compatibility with IE.
.show() - just remove the display:none; via el.style.display = "";. Btw, you might just want to remove the hidden class instead of overwriting it with an inline style :-)
Real vanilla for W3-compliant browsers:
[].each.call(document.getElementsByTagName('select'), function(select) {
select.addEventListener("change", function(e) {
if (this.value == "Other (please specify)") {
var inputs = this.parentNode.parentNode.querySelectorAll("input.hidden");
for (var i=0; i<inputs.length; i++)
inputs[i].classList.remove("hidden");
}
}, false);
});
This should work in older browsers, too:
(function(selects, handler) {
if (document.addEventListener)
for (var i=0; i<selects.length; i++)
selects[i].addEventListener("change", handler, false);
else
for (var i=0; i<selects.length; i++)
selects[i].attachEvent("onchange", handler);
})(document.getElementsByTagName('select'), function() {
if (this.value == "Other (please specify)") {
var inputs = this.parentNode.parentNode.getElementsByTagName("input");
for (var i=0; i<inputs.length; i++)
if (/\bhidden\b/.test(inputs[i].className))
inputs[i].style.display = "";
}
});