Background:

In this scenario assume technology is initially equal to our own, but a method is discovered to render material indestructible. The affected material is treated as requiring infinite or arbitrarily high amounts of energy to break any of its bonds whether they be nuclear or chemical (this does mean a previously radioactive material will no longer be able to decay).

Indestructible material can deform provided this wouldn't require breaking bonds or stretching them beyond what would have been possible for the starting material.

The process to make something indestructible costs hundreds of millions of dollars per cubic meter affected so answers should be limited to scenarios where using such an expensive material makes financial sense. Making an object indestructible involves placing it in a sealed reaction chamber and applying the Mcguffin effect to everything within, so you can't make only part of a contiguous object indestructible.

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

At the very least though I'd imagine there's great utility for power generation in taking advantage of this Mcguffin's ability to easily contain extreme pressures indefinitely (emitting energy through radiation, heat emitted by the vessel and light if the vessel is transparent).

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.

How could indestructible materials be used in power generation?

Energy storage.

If you can spin a flywheel to relativistic speeds on indestructable bearings using electromagnets (in vacuo), 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 miniscule ammount, 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 jouls 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.

Speculativley: 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.

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.

"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.

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.