I think it's usually no problem to get a doctor to order a CBC/chem/lipid profile, which helps keep an eye on liver/kidney function, electrolytes, blood cells, cholesterol, etc. I've been enjoying Mike Lustgarten's approach where he's aiming to optimize all of these for longevity (QS presentation and blog). It also seems like people have good luck getting doctors to order vitamin D testing.
Chris Masterjohn has a good series on which nutritional blood tests might be useful in different situations (series intro, glutathione, iron, copper, selenium, and zinc). It's nice because it may help you determine which of those might be justified for you.
I personally monitor some additional ones that might be harder to get covered by your doctor/insurance like prolactin (as a proxy for serotonin, estrogen, and thyroid), PTH (for calcium-handling) and CRP (although I think this is becoming more common to test if you have risk factors for heart attacks or stroke). I have a site about ordering your own bloodwork if there's any you'd like to test that your doctor doesn't want to order.
[edit: I hadn't seen Eric's reply when I posted, I'm in agreement that you're unlikely to get them to cover anything comprehensive]