The wider the gap between the exercise price and the fair market value the more AMT you will owe on the shares when you exercise. By making a speculative investment -- by exercising early -- you will owe minimal AMT at exercise time and more capital gains later when you sell.
There are very real tax advantages to exercising early.
If you really believe in the company and it's a rocket ship (some rockets explode mid-flight or stall though!) then I'd recommend buying some shares early on to hedge for the reason you state. But I probably would't early exercise 4 years of options on your first day at the company.
There are very real tax advantages to exercising early.