Is there a jQuery autogrow plugin for text fields?

Here's a plugin that'll do what you're after:

EDIT: I've fixed the plugin as per Mathias' comment. :)

See a demo here: http://jsfiddle.net/rRHzY

The plugin:

(function($){

    $.fn.autoGrowInput = function(o) {

        o = $.extend({
            maxWidth: 1000,
            minWidth: 0,
            comfortZone: 70
        }, o);

        this.filter('input:text').each(function(){

            var minWidth = o.minWidth || $(this).width(),
                val = '',
                input = $(this),
                testSubject = $('<tester/>').css({
                    position: 'absolute',
                    top: -9999,
                    left: -9999,
                    width: 'auto',
                    fontSize: input.css('fontSize'),
                    fontFamily: input.css('fontFamily'),
                    fontWeight: input.css('fontWeight'),
                    letterSpacing: input.css('letterSpacing'),
                    whiteSpace: 'nowrap'
                }),
                check = function() {

                    if (val === (val = input.val())) {return;}

                    // Enter new content into testSubject
                    var escaped = val.replace(/&/g, '&amp;').replace(/\s/g,'&nbsp;').replace(/</g, '&lt;').replace(/>/g, '&gt;');
                    testSubject.html(escaped);

                    // Calculate new width + whether to change
                    var testerWidth = testSubject.width(),
                        newWidth = (testerWidth + o.comfortZone) >= minWidth ? testerWidth + o.comfortZone : minWidth,
                        currentWidth = input.width(),
                        isValidWidthChange = (newWidth < currentWidth && newWidth >= minWidth)
                                             || (newWidth > minWidth && newWidth < o.maxWidth);

                    // Animate width
                    if (isValidWidthChange) {
                        input.width(newWidth);
                    }

                };

            testSubject.insertAfter(input);

            $(this).bind('keyup keydown blur update', check);

        });

        return this;

    };

})(jQuery);

If you want the textbox to grow when the string within it extends past its width, maybe something like this would work for you... It detects the size attribute of the textbox. If the length of the string goes over that attribute, it extends the textbox to the length of the string on keyup.

In the below script, "#test" is a textbox ID.

<script language="javascript" type="text/javascript">
$(document).ready(function(){
    $("#test").keyup(function(){
        if($("#test").attr("size") < $("#test").val().length){
            size = $("#test").val().length;
            $("#test").attr("size",size);
        }
    })
});
</script>

Funny enough in IE overflow: visible is taken very seriously. You can achieve this effect by applying overflow: visible on your input elements. Not sure if any similar CSS tricks exist for modern browsers.