Is there an application that will tell me the compass bearing of what direction my laptop is facing?
First of all, as user25866 points out, there's no way to find out without extra hardware.
Secondly, even a GPS receivers will not tell you this, as GPS only gives you your position (as in "dot on a map"), but not your orientation. AFAIK, this is a fundamental limitation of GPS.
Getting a heading from GPS is only possible in two ways:
There are "digital compasses" for sale, which basically have a real magnetic compass inside which they read. Those I found are for industrial applications, though; dunno if there's one that you can connect to a laptop.
Also note that e.g. the Vaio-P-Series laptops have a built-in digital compass (in addition to GPS). Maybe check those out...
No. Finding your orientation in the physical world requires hardware. For example, even if you have software to examine the position of the stars, you still need a camera to see them, which you might as well just have a GPS.