Compile a LaTeX document into a PNG image that's as short as possible

You can use the standalone class for this. It loads the preview package automatically to crop the resulting PDF to the content. This makes the usage of pdfcrop unnecessary.

Simply exchange the article class with standalone. (It uses article internally but another class can be specified using the class option.) Note that since v1.0 the default option has been changed from preview to crop. The latter is better for pictures etc. but doesn't support line breaks. Either select preview manually or use the varwidth option.

\documentclass[preview]{standalone}
\begin{document}
Hello. This is a test.
\begin{equation}
L = 2
\end{equation}
\end{document}

There is a border class option which sets the border around the content (the default is 0.5bp). This option excepts either one (border for all sides), two (left/right, top/bottom) or four values (left, bottom, right, top).

To convert it to a PNG I recommend to use the convert command of Image Magick:

pdflatex file
convert -density 300 file.pdf -quality 90 file.png

Here the density is 300 DPI which can be adapted to your needs. The quality setting selects the compression level and other things and 90 is AFAIK the optimum.

You can also select the DPI resolution for X and Y separately and also resize the resulting image, e.g.:

convert -density 600x600 file.pdf -quality 90 -resize 1080x800 file.png

---

Update 2011/12/21:

The new version 1.0 standalone now has the ability to call the above command line (and others) automatically, e.g.:

\documentclass[convert={density=300,size=1080x800,outext=.png}]{standalone}

or simply (using default setting 300dpi, no resizing, PNG):

\documentclass[convert]{standalone}

This needs the -shell-escape compiler option to allow the execution of the conversion program from within the LaTeX document.

One thing that's easy to miss is page numbers. The page number restricts the height of the final image so it's best to leave it out. An easy way to do that is to use the empty page style.

What I do when doing images for this place is to have a document a bit like:

\documentclass{article}
\thispagestyle{empty}
\begin{document}
Hello. This is a test.
\begin{equation}
L = 2
\end{equation}
\end{document}

Then run pdflatex on it to get a PDF; next run pdfcrop (comes with TeXLive) to make it as small as possible; finally convert it to PNG using the NetPBM library tools. (This is on a Unix machine.) So my workflow is:

pdflatex document.tex
pdfcrop document.pdf
pdftoppm document-crop.pdf|pnmtopng > document.png

et voila:

If you have a dvi file file.dvi, running dvipng -T tight file.dvi will produce a png with the image automatically cropped as much as possible. (You might also want to set the output resolution using the -D flag, as in dvipng -T tight -D 150 file.dvi for 150 dots per inch.)

As Andrew points out in his answer, getting rid of the page numbers is a good idea.