What material is used to make the hot sections of jet engines?
Short answer


Fan: Aluminum, titanium, or stainless steel
Compressor: Nickel-, cobalt-, or iron-based alloys. Additive are aluminum and/or titanium, chromium, as well as rare earth elements like yttrium.
Combustion chamber: Superalloys with refractory metals such as tungsten, molybdenum, niobium, tantalum. Ceramics and ceramic-metal mixes.
Turbine: Nickel-based superalloy, outside air is circulated through channels inside of the turbine blades. For lower pressure turbine blades iron-based superalloy or even stainless steel. The metals used for turbine blades are often grown as a single crystal.
Exhaust nozzle: Nickel Inconel and stainless steel alloys.
Casing: Aluminum or polymer matrix materials.


The high-pressure turbine is, with the combustion chamber, very hot (more than the exhaust nozzle). You may read this Wikipedia article which has a short list of materials used for turbine blades (with alloys names and cooling techniques).



Engine structure (turbofan)


(source: Wikipedia)

An actual engine looks like that (CFM56 turbofan). It's easy to map the detailed elements to the simplified schematic. The fan pushes about one ton of cold air in the engine every second in high power turbofans. This is why temperatures and pressures can be so large after compression and after combustion.

All quotes are from Nasa Guide To Engines, 2007.


  In 1903, the Wright brothers built an aluminum
  block engine because of its light weight compared to cast iron. Its
  melting point of 660 °C was well above the engine’s operating
  temperature, so for them it was a good choice. However, aluminum could
  not be used in the hotter parts of a turbine engine where temperatures
  reach of 1800 °C or more because it would melt.


Fan


  This typically does
  not get very hot (&lt;150 °C) so aluminum, titanium, or stainless steel
  are all suitable for the fan blades. Most engines use titanium because
  it has a high strength-to-weight ratio, is corrosion and fatigue
  resistant, and would be able to withstand the impact of a bird strike.


Fan example (with a fan blade missing after the engine blew apart).

Compressor section


  The pressure of the air can be raised up to 30 times and the
  temperature, depending on the number of stages in the compressor, can
  rise to 1000 °C. Here the materials must have high strength at high
  temperatures; must resist fatigue, cracking, and oxidation; and also
  must resist “creep.” Creep is the tendency of a material to slowly
  change shape when stressed at high temperature. Since no single metal
  would have all the desired properties, an alloy (a mixture of metals)
  is used. Very high- temperature alloys are called superalloys and are
  generally nickel-, cobalt-, or iron-based alloys. Aluminum and/or
  titanium are added for strength, and chromium, as well as rare earth
  elements like yttrium, are added to improve corrosion resistance.


Compressor example (high pressure rotors)

Combustion chamber


  Temperatures can exceed 1800 °C and again superalloys are used, but
  without the titanium or aluminum for strength because there are no
  moving parts. Instead, refractory metals are often added to a
  superalloy. These are metals of unusually high resistance to heat,
  corrosion, and wear such as tungsten, molybdenum, niobium, tantalum,
  and rhenium. They are used in alloys and not as pure metals because
  they are among the densest of all the elements, a negative property
  when it comes to aircraft that need to keep weight to a minimum.
  Ceramics and ceramic-metal mixes are also used here because of their
  high heat resistance. We are familiar with pottery, tile, crucibles,
  and fire bricks as types of ceramics. They have very high melting
  points and don’t require the cooling systems like those needed to keep
  metals from melting so they make for lighter, less complicated engine
  parts. The down side is that they tend to fracture under stress, so
  engineers seek to create new ceramics composites that incorporate
  other materials to improve properties.


Combustion chamber example

Turbine


High-pressure turbine blade (source: Wikipedia)


  The first set of turbine blades are in the highest pressure, hottest
  part of the gas flow and are generally made of nickel-based superalloy
  or ceramic blades. Unheated outside air is circulated through channels
  inside of the turbine blades to keep them from melting in this extreme
  environment. Further down the engine lower pressure turbine blades
  often sit. Since the gases have somewhat cooled by this point, the
  blades can be made of iron-based superalloy or even stainless steel.
  It is interesting to note that for strength, the metals used for
  turbine blades are often grown as a single crystal. A close look at
  most metals and alloys show that they are composed of crystals (also
  called “grains”), and the places where the crystals meet are called
  grain boundaries. A material is weaker at the grain boundaries than
  within the grains—especially at high temperatures— so turbine blades
  fashioned from metal formed as a single grain (no boundaries) are
  stronger.


See single crystal material (known as monocrystalline material in electronics) .

Turbine example (high pressure stator)

Exhaust


Exhaust from Boeing 787 engine (GEnx, one of the world's largest jet engine - Source)

From FAA Aviation Maintenance Technician Handbook - FAA-8083-30:


  Inconel and stainless steel alloys. The Inconel [nickel-chromium-iron]
  alloys are frequently used in turbine engines because of their ability
  to maintain their strength and corrosion resistance under extremely
  high temperature conditions.


See Inconel details.

Casing


  Although it need not withstand high temperatures like the core of the
  turbine, the materials here need to be strong enough that if a blade
  were to break off it would be contained within the casing and not
  enter the wing or cabin of the aircraft and cause further damage.
  Aluminum or some polymer matrix materials are used as engine casings.