It's far from a full solution, but the XY Chart Labeler might help you get part of the way there: http://www.appspro.com/Utilities/ChartLabeler.htm - I find it indispensable for my bubble and scatter charts.

Chris, it's easy enough to mimic the NYT chart with your data in Excel. A couple of custom series to represent labels (e.g. site name and terms), some custom formatting and you've got a decent representation. The problem of labels in this case I think is the relative smoothness of the data. the NYT data ranges from 1 to 21 which compared to your sample data which runs 0-100% is a relatively tight grouping so the bubble scales are much more similar. In other words, it was a lot easier for the NYT team to fit labels on their bubbles than it is to do on your data. I'll post a graphic later on my blog but, I think the acceptable solution is to create a size legend series...

Scott, in my attempt, I did use x/y chart labeler (a must-have for excel) but that was for custom row and column labels, I don't think it will really solve for getting the labels onto the buttons.

Hey Chris,
I was about half way through writing the blog post and encountered a lovely BSOD. Too tired to start the post over tonight so here is the URL to the screenshot of my solution:
<img src="http://blog.instantcognition.com/wp-content/uploads/2009/01/juice_concentrate_bubble.png" width="350px" height="287px" border="0" alt="Clint's Excel Bubble Chart Solution" title="Clint's Excel Bubble Chart Solution"/>

Wow, didn't think that would actually work.
In any case, to follow up on my original comment about the smoothness of the data...
The area of the bubbles in the NYT graphic scale from 1 to 21 which means the diameter has to scale 21x from smallest to largest. For the concentrate data the diameter scales from .5% to 100% - a 200x increase so it's much harder to make data labels fit into/onto the bubbles and be readable and preserve the integrity of the graphic itself

one last note ... don't do math when you're tired. As the above shows (still true, math is just wrong)

Just wanting to throw SAS' name in the hat, as another software that can do this kind of "bubble table" graph (with a bit of programming).

Here's an example of the output. I'm using gif here, which everyone should be able to easily view. SAS can also output other formats such as svg and 1200-dpi tiff, which make the bubbles look smoother (more "publication quality").

http://robslink.com/SAS/democd33/death_odds.gif

I just posted an Excel version of the chart based on the UDF I created.
Here : http://sparklines-excel.blogspot.com/

I put this together in Tableau:
http://tinyurl.com/9skr3h

Wasn't too sure what your colour was measuring, so I based it on size.

I tried it with text labels, but the text was too big relative to the bubbles. And I couldn't force the size variance to be any bigger. I would have liked to be able to make the biggest bubble bigger.

However, I think it's okay for 5 minute's work!

Andy,
yes, it's much simpler in Tableau and while I like your color as value indicator, you could just as easily have the values displayed next to the bubbles instead of 'on' them and Tableau keeps it clean.