This 40 AWG nichrome needs at least 8V @1A to get red hot. My 5V power supply @3A will get it hot, but only just. My variable 1A supply gets it red hot at 8V. I need to put this in a fixture with different lengths for testing.
Ian's tests suggests that steel wire gets hotter at lower amps. So nichrome's main benefit is that is withstands repeated abuse.
The temperature a given wire reaches depends only on the current, not the voltage.
The situation is best analyzed by considering a unit length of wire.
For our unit length, the power dissipated is given by the formula:
W = I²R
Where I is the current, R is the resistance per unit length (at the operating temperature), and W is the power dissipated per unit length.
We can see that for any unit length of wire, the power is proportional to the square of the current and directly proportional to the resistance.
Now the resistance per unit length is proportional to the cross-sectional area of the wire. (And temperature: resistance goes up with temperature, but if we are designing for a particular temperature we can assume the resistance to be the value that applies at our working design temperature.)
The cross-sectional area of the wire in the AWG system doubles with three steps in wire gauge. So for example, going from 40 AWG to 43 AWG would double the resistance per unit length.
So where does voltage come in? This depends on the length of the wire. If you halve the wire length you halve the total resistance and therefore halve the voltage required to obtain the same current.
Suppose you need 8 V to make 1 ft of wire get red hot. Then you need 4 V to make 6 in of wire get red hot, 2 V for 3 in, and so on.
Combining all these rules together allows you to determine the wire length, the wire diameter, the required current, and the required voltage to make your wire perform as desired. (You can assume that wires of different but similar thickness, within reason, will reach approximately the same temperature when subjected to the same amount of heat dissipation per unit length.)