Lowest voltage needed to turn on a FET?
This does not sound like a good fit for a FET since you only have 1.8V drive available.  FETs that switch natively with such a low voltage aren't likely to withstand 40V drain to source.

A bipolar only needs one diode drop to put enough current thru the base to turn it on well.  Getting one to withstand 40V with good gain is easy.  You didn't say how much current you need at 40V and what the inductor saturation current is, so let's say the switch needs to handle 500 mA.  You can easily find a NPN that can do 40V with a minimum guaranteed gain of 50, so that means you need 10 mA base current.  Let's say the B-E drop is 700mV, so the drop on the resistor is 1.1V.  1.1V / 10mA = 110&Omega;  As long as the processor can reliably provide 10mA out, that's all you need.

What power voltage are you boosting from?  If its the 1.8V, then the turn on voltage of the NPN will be a significant fraction of that.  Another possibility is to make a small analog booster to tun the micro off of 5V or so, then have it drive a FET directly.  FETs that turn on well with 5V on the gate and can withstand 40V will be a little less scarce.  That FET can still switch the 1.8V to make the 40V, but will get switched from the intermediate 5V supply.