Easy Peasy. Fusion reactors.

The primary challenge involved with fusion power is maintaining containment, which is a big challenge given the pressures and temperatures involved. I've included a link below, but here's the important bit:

"Not only will the neutrons deposit energy in the blanket material, but their impact will convert atoms in the wall and blanket into radioactive forms. Materials will be needed that can extract heat effectively while surviving the neutron-induced structural weakening for extended periods of time."

http://www.engineeringchallenges.org/challenges/fusion.aspx

The TLDR is that currently a PHYSICAL containment solution is impossible, requiring magnetic solutions that suck up most, if not ALL of the power being generated. Your Macguffin would solve this neatly, allowing a simple machined or cast sphere to be turned into a perfect containment vessel for a fusion powerplant of pretty much any size you need.

EDIT: Now that I'm thinking about it, it'd be a good solution for FISSION reactors as well, since a reactor vessel macguffin'd in the manner you describe wouldn't lose containment in a runaway nuclear reaction. The core could still melt down, but it'd stay in the reactor vessel. Your reactor would be destroyed, but it couldn't irradiate the entire powerplant ala Chernobyl or Fukushima.

Nuclear pressure containment is a good method.

Nukes have to be held together to make fission continue for as long as possible. If you hold 20 critical masses together for a full second, you'd generate the largest nuclear explosion ever made by humans.

With indestructible materials, you could hold them together for an hour. At those high energies, there are all sorts of effects that release even more energy.

Make a box out of indestructinum. Put a nuclear bomb in it. Detonate and let it build up fusion-capable pressure. Slowly vent it out to generate power. If your material conducts heat, put it in a very effective cooling system and generate power reactor-style.

Alternatively, vent it out quickly in the direction of someone rich until they give you what you want.

How could indestructible materials be used in power generation?

Energy storage.

If you can spin a flywheel to relativistic speeds on indestructible bearings using electromagnets (in vacuum), then you can use that flywheel as a lossless energy storage device.

_{Wikipedia 2019 - CCSA License}

The energy density would be infinite (or limited by the unspecified arbitrary high amounts of energy in the question) - thus you would need a microscopic minuscule amount, a nano-flywheel mounted on gimbals - radically reducing the price per flywheel and opening it up to mass marketing, totally outclassing all battery tech available today.

Not only the obvious solution to the supply and demand issues with windpower, but for vehicles - cars/planes, phones, power-tools, toys, mobile phones and of course space exploration.

Infinite energy storage in the size of a grain of sand.

Miniature Tactical Nuke:

Of course, this section is about political power generation.

To release all that energy in one instant - perhaps an object charged with just below the threshold of it's (unspecified arbitrary potential energy) capacity, could be placed near an enemy stronghold and fed that last few joules of energy to tip it over the edge, that's the dark side, someone will find a way to weaponise it for sure, if not the leader of some isolationist sanctioned state, then a disaffected teenager.

Power of a civilisation through time travel.

Speculatively: Also it would have potential to enable time travel or at least the potential to send messages back in time as it would exhibit frame dragging. For a few hints on how this could be of tactical use see this answer to another question.

For a more nerdy approach, you could build indestructible turbines.

(Disclaimer: My memories of thermodynamics are fading in the mists of time, so feel free to blast me in the comments if I'm wrong).

I remember that in thermal power generation (where water is heathed into steam, whose energy is used to move a turbine), they were forced to limit the calor of the steam in output from the turbine, thus reducing the efficiency (basically, the colder the exiting steam, the better the efficiency).

The reason was that if the water steam was allowed to cool too much, it would condensate and create water droplets that would move so fast to act as bullets, damaging the turbine.

But an indestructible turbine could easily withstand this scenario, thus allowing for exploiting all the energy of the steam and generating more power.

Of course it is necessary to evaluate if the increase in efficiency is enough to compensate for the higher cost of the indestructible turbine.

Construction

A thin, unbreakable wire added to dam wall construction, would allow dams to be built cheaper, thinner, stronger and higher.

A unbreakable foil added to the overflow means it would never wear out and need replacing

Large dams for energy generation cost from twenty to thirty billion dollars so an extra cost of a couple of hundred million to make it unbreakable would easily be offset by the less concrete and steel needed not to mention to maintenance costs down the track plus the safety of an unbreakable wall.

If you want next level power generation may I present

The Space Elevator

A thin unbreakable wire running to an orbital platform with a twin on the moon, would allow the efficient harvest of He3 from the lunar surface which could power fusion reactors around the world.

While the fusion power example is the most practical, you can connect a massive solar collector in orbit to the ground via an indestructible fibre-optic cable, and send a laser pulse down the cable to a boiler powering a steam turbine. As a bonus, since the cable is indestructible, you could use it to tether a space elevator, and get cheap rides to orbit.

Sometimes the smallest thing has the largest impact

Do you know how much wire you can extrude from a cubic meter of copper when you can trust it to be indestructible?¹

Indestructible insulating enamel + indestructible conductive wire = the perfect transformer/motor/generator.

When was the last time you opened a power supply, motor housing, generator, or anything using inductive windings, and found the transformer/motor/coil burned out. For me, it was last week (literally, it was last week). If you could make both the wire used in the windings and the enamel used to coat the wires indestructible, what you would have is the perfect transformer/motor/generator.

Yeah, but this stuff is expensive

Which is why it would make sense for large items, like turbine-style power generators where the limit to the electricity you're generating is suddenly the mechanical stress limits of the linkages and not the heat-generating characteristics of the coils. Better still, indestructible windings and enamel means you can make the coils incredibly dense — and as coil density increases, so does power output. Your efficiency might actually approach unity. Imagine a wire that is no longer a fuse if too much power is put through it. There is no longer too much power, the limitation is literally the speed electrons can be induced to move through the wire.

And if you expand to power utilization, the applications become … impressive

Miniature motors that can turn the propellers on a submarine? Dock 6. Full-size motors that push submarines at tsunami-creating speeds? Dock 2. Car alternators the size of your thumb? Aisle 14. A Dremel the size of a pencil? Aisle 1. An electric car that actually works climbing the Rockies? The display arrives next week. A residential wind turbine that actually powers an entire house? We have on the roof, you can see it as you enter the building.

The process may be expensive, but the material requirements (in terms of how much you need) drop like a rock when you can trust the wire and enamel to be indestructible. The process of making things indestructible would benefit almost any application at any price. A steam boiler the size of a Buick enjoying such high pressure that it can pull a mile-long train? On display by the front counter.

Disclaimer: at hundreds of millions of dollars per-cubic-meter there it is unlikely that any application is worth it. Unless you can jack the price through the roof, the cost recovery time at that price relegates the material to use (not necessarily power generation) in remote locations (like space) where repair costs even more. A spaceship hull would be worth that price. I frankly can't imagine any power generation/utilization solution that ever would. Not even fusion. The cost of using something less capable would be so much more economical that such a solution would only happen as a test, never a commercial solution. So, a frame challenge concerning the price.

_{¹ A cubic meter of copper weighs 8,930 Kg. 40 Gauge wire weighs 0.04454 grams/meter for 200,490.6 Kilometers of wire. That's enough wire to wrap the equator 5 times.² And you might be able to use thinner wire than that. It's a lot of honking wire.}

_{² Of course, the wire is indestructible. If you wrapped the equator just once and tied the two ends to space ships, assuming a reasonable amount of thrust, could you garrote the world in half? It gets the mind wondering, doesn't it?}

"So how could these aforementioned indestructible materials be used in conjunction with existing or near future technology to improve power generation?"

Well, if you could make copper indestructible then you could use it as I mentioned in my comment. Simply dig a very deep hole and place a copper rod in it. The heat at the bottom of the hole would conduct through the rod to boil water at ground level. The boiling water would be used in a convention steam turbine and BAM nearly infinite free and clean energy. The only reason we don't already do this is because copper would melt at the temperatures needed to get enough heat conducting through the rod to boil water on the other end. That and it would be very hard to dig a hole that deep because all the drill bits would melt but since we can make indestructible drill bits, it should be no problem... heck, we could reach the core with indestructible material.

Jet engine turbine blades

The limit in turbine efficiency (and why jet engines keep getting better and higher-bypass, but sloooowly) is the thermal limits of the first-stage (right behind the combustor) turbine blades. Evolution is waiting on new material-ally tech and more extreme methods of cooling(already pretty extreme and energy-robbing).

If you can make the first two stages out of indetructium, as well as a few combustor-area components that would be hard to airstream-cool, you can keep pushing up the efficiency of the engines. Now you have 20:1 or 30:1 bypass. Turboprop sub-chaser maritime patrol aircraft that can stay on station for 48 hours. Over on the ship-propulsion or terrestrial power generation side, you have ships with more range between tankers, and power plants with lower smog and cheaper power from natural gas and petroleum. It would be the deathknell of coal.

At a minimum, if you replaced all your ball bearings with indestructible bearings, you'd be a long way towards better energy production.

Many generators in the energy industry have to be tore down periodically to have their bearings and fins/rotors replaced. Never having to do this will save that cost, including labor and the energy production to cover the "down" generator.

In fact, making the entire generator out of these indestructible materials would be a major boost. Hydroelectric dams could run at any speed, same with the huge windmills. (Ever seen a windmill with a failed break mechanism break up? YouTube that if you want to cringe.)

As much as the new forms of energy production suggested by other answers would help, simply replacing the mechanisms of the existing system would help considerably. This might be a lower investment level to get poor countries more power.