Short answer: no.
50% longer answer: yes.
But seriously, how hot the wire will get for a given power input will depend on how well it's insulated. If the insulation is perfect, then there is theoretically no limit to how hot you can make it. Realistically, either at some temperature the heat radiated to the environment equals the power input, and it stops getting hotter, or the wire melts.
I'm not entirely sure what "4.7mW at 47% efficiency" means. 47% of what? Is 4.7mW the power output, or input of the power supply? All that's really relevant in the power output of your power supply, because the power input to the wire will equal exactly this, and 100% of this electrical power will be converted to (what else?) heat.
Anyhow, let's say 4.7mW was the power you put into the wire. How power translates to temperature rise for a particular component is usually expressed as an absolute thermal resistance in units \$K/W\$, or equivalently, \$^\circ C/W\$. It expresses how much temperature rises per unit power input.
So let's say you want a temperature of \$750^\circ F \approx 400^\circ C\$, and ambient temperature is \$20^\circ C\$, thus you need to raise the temperature \$ 400^\circ C - 20^\circ C = 380^\circ C\$, and you need to do it with \$4.7mW\$. The thermal resistance must then be:
$$ \frac{380^\circ C}{4.7mW} \approx 80800 ^\circ C / W $$
or more. That's not physically impossible, but it's some pretty serious insulation. For comparison, a typical thermal resistance to ambient for a typical TO-220 package, which is approximately the same size as you describe, is \$34 ^\circ C / W\$. This table of thermal resistances of common packages from Linear Technology may be helpful to put those numbers in perspective.
You also asked "how long", and again the answer is complex, and will depend on the environment and insulation. The ultimate speed of heating will be limited by heat capacity of your heating element and the rate of energy input (power). This will be offset by the rate at which heat is lost to the environment. But, having already established that you need some pretty incredible insulation to achieve the target temperature in your case, we can skip the rigorous analysis and intuitively know that the high temperature is reached by integrating the power over a long time to accumulate enough energy, and say simply the time required to reach the target temperature is "a long while".
This mostly depends on what your powering the resistance wire with. The voltage going in, combined with the resistance of the wire will dictate the current flow and therefore the power dissipated in the resistance wire. All power dissipated in the resistance wire is useful heat, to heat up your materials (chemicals/shisha etc).
So the actual resistance is not relevant if you have absolute control over the voltage being fed in. As a rough indicator, a 10W heater with a 12V voltage source would need a total resistance of 14.4Ohm (Pd = V^2 / R), which with this wire would need to be 3.42m long.
How much thermal power you produce (Pd) will then dictate the temperature, depending on the materials you are heating. Thermal energy will be needed to bring up the materials to the desired temperature, and less energy will be required to keep the materials at the required temperature (due to heat escaping to the environment).
To keep a constant temperature, you need to reach a state of equilibrium where thermal energy added to the system matches thermal energy being lost. Heat loss will increase as the system temperature increases, so for any given heater system it will naturally reach a maximum temperature. However, any change to the system will change the maximum temperature (remove materials, the wire may overheat).
In short, you can make a heater system that does not have a controller, but the required power fed in is highly dependent on the system (what materials you are heating, how much etc). Your best bet is to use a lab power supply, in current control mode, and slower creep up the current until it heats up and stabilizes at the required temperature. You can do calculations to estimate heat loss, but they may not be much more useful than empirical testing.
As a rough guide for somewhere to start, make an educated guess as to the rough order of magnitude of power needed. This device heats about 0.5sq m of soil to around 30C, and is rated at 27W. So we could infer that for temperatures of 30-50C, over an area of about an A4 piece of paper to 0.5sq m, 10-30W would do the trick. This is assuming that the wire is heating something else up first (some medium like sand). This is a Fermi style estimation, to give a rough idea or scales. If you start testing with the ability to generate 10-30W of heat from your wire, you can then ramp up your power supply across the range to see what works.
If you do make a heater without a controller, you still need some kind of safety trip to stop it overheating. If you do some googling, you can find various ways to do this. The easiest is probably a simple thermal fuse in series with the resistance wire, that will disconnect power to the resistance wire if it heats beyond a set temperature.
Best Answer
The first thing you need to figure out is how much power per unit length it takes to maintain the wire at 43 °C (110 °F). That depends on things you haven't told us. The ambient temperature and thermal conductivity of whatever the wire is touching will make a large difference.
Calculating the power required to maintain a particular wire temperature is probably not possible since too many things will be unknown. The best method to find the answer is to try it.
Get some wire, connect it to a lab supply, and see how much current it takes to keep it at the desired temperature. This doesn't need to be nichrome. Just about any wire can handle 43 °C. When find the setting that yields the desired result, record the current and the voltage drop across a known length of wire. The current times the voltage is the power being put into that wire.
From the power into a known length, you find the total power into your desired length. The power into a resistance is:
W = V2 / Ω
where W is the power in watts, V the voltage across the resistance, and Ω the resistance in Ohms. If you are going to apply a fixed voltage, like your 3.7 V battery, then flip this around to find the resistance:
Ω = V2 / W
Now you look at wire resistance tables to find some that has the desired resistance over that length.