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Say that a company has a monopoly upon the water market. They are the only company that can provide said resource, and they can choose to set the price however they want.

Not true. What is stopping me from creating a company that also sells water?

I covered that. A monopoly means that they are the exclusive owners of a product in a market, and even if the were no longer a monopoly (by your creating a company that also sells water), they would be easily capable of bankrupting you in any number of ways (many of them legal, such as buying out all of your stock and making themselves the owners of "your" company). Also, how are you going to get the water to make this company... especially when the monopoly has already employs those who collect the water to be sold themselves?

They will thus attempt to maximize their profits... and they will sell only the worst "equal parts muck and water" to those in the worst financial situations (likely for prices that they will only be able to pay for by sacrificing their food, heating, shelter, property, etc), and the somewhat less 'dirty' (maybe toxins, maybe it runs in with the sewage, perhaps the water has become so overpriced that there is no longer an operating sewage for the middle-poor classes, thereby the smaller plumbing businesses have all been declared bankrupt) water is sold at exorbitant prices to the middle and poor classes... to the point where it may be simply more economical to drink the "mudwater" sold to the poorest of the poor.

Your example, whilst quaint, does not happen in the real world. Why? Because as soon as quality goes down, price goes up, etc people start to notice, and complain. If the company doesn't sort out the problem, other companies will be set up to match the new demand for cleaner, cheaper, water.

To take a more realistic example, consider the monopoly on transporting goods in the 18th century in England. Back then, canals were used to transport goods, as barges could support heavy loads and transport items safely. The canal business monopolised because of this fact; everyone wanted their goods transported, and there was no better alternative. Then...the train was invented. If you want to stop a monopoly, do what they do, but better.

It doesn't matter if people complain. There is absolutely nothing that can be done (outside of government action) to compete for (perhaps not even obtain and/or sell) the resource that a monopoly is based around.

My example may have been "quaint"... but the disgusting thing is that worse things have happened. Remember the East India Trading Company? The things that Upton Sinclair asserted in The Jungle were for the most part true. Exploitation on a mass scale is what you get with no regulation... just look at the alaskan salmon fishing industry http://en.wikipedia.org/wiki/Alaska_salmon_fishery . There are damned good reasons why the government should be a part of the market, or at least why the market needs regulations.

What... except that the one with the monopoly controls all of the market already, and can afford to buyout all of the shares of your upstart company?

I don't know of many upstart companies that float on the stock market.

Or perhaps the monopoly has already struck exclusive deals with suppliers who are also monopolies... and the price to beat them out is beyond that which you can afford?

Favouritism isn't an attribute of free market capitalism. The keyword here is "free".

Did I say favoritism? No. I suggested that the company would offer its services to the highest bidder (because they are being paid more by them)... and that any upstart trying to beat the monopoly on this front must firstly have as much or more money than the company with the monopoly.

There is simply no way a fresh upstart can compete with a monopoly... not unless they are both able to capable of supplying all the things the now former monopoly is capable of supplying and able to compete with their prices and connections.

Ignoring the connections aspect since it is irrelevant (see above), you've mentioned previously that monopolies sell a low quality product for a high price. All a competitor needs to do then, is to sell the same quality product for a slightly lower price, and they've caused a problem for the monopoly company. Or, make a better quality product for the same price. Either way, if a lot of new businesses started doing this, you'd see a rapid change in the monopoly structure, as it tried to lower its prices or make a better product...both of which I can only see as good things.

Or rather, you'd see a lot of those small and new businesses bought out or go bankrupt as they attempt to compete with the industry giant.

Not to mention that if this is a truly free market... assassins and armies are services too...

Evidently you don't know a thing about the free market. The free market is not above the laws of the country. If a monopoly paid an assassin to kill competitors, they would be committing a crime. Same if they created an army.

That is imposing government regulations (laws) upon the market. Hence the market is not free. I think I understood the concept of a 'free market' well enough, thank you very much. I do believe that the http://en.wikipedia.org/wiki/Ferengi or http://www.wowwiki.com/Goblin or other such trade-princes governments are perfect examples of a free market... although the Ferengie actually do have a few "rules".