make html text input field grow as I type?

If you set the span to display: inline-block, automatic horizontal and vertical resizing works very well:

<span contenteditable="true" 
      style="display: inline-block;
             border: solid 1px black;
             min-width: 50px; 
             max-width: 200px">
</span>

Here is an example with only CSS and Content Editable:

jsFiddle Example

CSS

span 
{
    border: solid 1px black;
}
div 
{
    max-width: 200px;   
}

HTML

<div>
    <span contenteditable="true">sdfsd</span>
</div>

Important note regarding contenteditable

Making an HTML element contenteditable lets users paste copied HTML elements inside of this element. This may not be ideal for your use case, so keep that in mind when choosing to use it.

How about programmatically modifying the size attribute on the input?

Semantically (imo), this solution is better than the accepted solution because it still uses input fields for user input but it does introduce a little bit of jQuery. Soundcloud does something similar to this for their tagging.

<input size="1" />

$('input').on('keydown', function(evt) {
    var $this = $(this),
        size = parseInt($this.attr('size'), 10),
        isValidKey = (evt.which >= 65 && evt.which <= 90) || // a-zA-Z
                     (evt.which >= 48 && evt.which <= 57) || // 0-9
                     evt.which === 32;

    if ( evt.which === 8 && size > 0 ) {
        // backspace
        $this.attr('size', size - 1);
    } else if ( isValidKey ) {
        // all other keystrokes
        $this.attr('size', size + 1);
    }
});

http://jsfiddle.net/Vu9ZT/

I just wrote this for you, I hope you like it :) No guarantees that it's cross-browser, but I think it is :)

(function(){
    var min = 100, max = 300, pad_right = 5, input = document.getElementById('adjinput');

    input.style.width = min+'px';
    input.onkeypress = input.onkeydown = input.onkeyup = function(){
        var input = this;
        setTimeout(function(){
            var tmp = document.createElement('div');
            tmp.style.padding = '0';
            if(getComputedStyle)
                tmp.style.cssText = getComputedStyle(input, null).cssText;
            if(input.currentStyle)
                tmp.style.cssText = input.currentStyle.cssText;
            tmp.style.width = '';
            tmp.style.position = 'absolute';
            tmp.innerHTML = input.value.replace(/&/g, "&")
                                       .replace(/</g, "&lt;")
                                       .replace(/>/g, "&gt;")
                                       .replace(/"/g, "&quot;")
                                       .replace(/'/g, "&#039;")
                                       .replace(/ /g, '&nbsp;');
            input.parentNode.appendChild(tmp);
            var width = tmp.clientWidth+pad_right+1;
            tmp.parentNode.removeChild(tmp);
            if(min <= width && width <= max)
                input.style.width = width+'px';
        }, 1);
    }
})();

JSFiddle