Since vector drawings are scalable infinitely, I should be able to save my drawings at any DPI I wish. How do I do that with Illustrator CS5?
There ain't no such thing as "DPI" in an Illustrator document, except when you're referring to applied raster effects. There are no dots, so there can't be any "dots per inch". The maximum document dimension of 227.54 inches (actually 16383 points) is a constraint in the way that Illustrator holds the dimensions internally, a legacy of its original purpose: to enable a user to create Postscript without having to write code.
To get some dots, save your document as a PDF, then open the PDF with Photoshop. In the Open dialog, you can specify the PPI you want Photoshop to rasterize it at, and within the limits of your computer's memory and the space Photoshop has to work in, you can, indeed, make it whatever resolution you like.