Economists usually ascribe three properties to money.  These are:

Medium of exchange
This is not a unique property of money, as many other goods are used for transactions, though with varying frequency.
Store of value
Everything is a store of value to some degree.  If you put a ton of lumber in a warehouse, it will be a ton of lumber next year.
Unit of account
This means that prices are quoted in terms of money.  This is the only property of the three that only applies to money.

Because common goods are good or bad stores of value, this property is well-investigated by itself, without reference to money.  Investigating the unit of account could happen somewhere else.  Here, I'll explore the gradual property of being a medium of exchange.

A good that can be used in transactions right now and in large quantities, without needing to lower the ask price, is said to be liquid.  For example, if your plumbing bursts and you must sell something right now to pay for fixing it, furniture is a bad choice.  You would have to ask a very low price for someone to buy it immediately.  Thus furniture is not liquid.

On the other hand, if you have stocks, bonds, forex or cash, you can sell that immediately without a loss.  Indeed, the plumber will certainly do the work for your cash or bank deposit directly—and he might take forex in payment, though that's not very common.  These assets are liquid.

If you think you might need to put out a fire, you want to have a fire extinguisher, and are willing to pay for it.  If you think you might need to sell something on short notice without loss, you want to have some liquid assets, and are willing to pay for it.  Even if you never end up using either the fire extinguisher or the liquidity, you must pay for its price, because somebody else would have wanted to have it.

As you see, liquidity is a valuable property.  An asset that is more liquid than another will have some liquidity premium in its price over that of the other.

More generally, the prices of securities will include a liquidity premium.  After all, if two securities have exactly the same credit risk, term risk, and fundamental value at expiry, but one is more liquid than the other, then the first will be a better deal at the same price.  Even if you don't think you will need its liquidity, if it's being offered for free, why not take it?

Measuring liquidity

How much premium?  Unfortunately, we don't know.  Liquidity is like the view from a room.  You can only sell the room and the view together, not each separately.

Although in a few cases we can measure the difference, because there are identical rooms with and without views.  The difference in interest rates between a demand deposit and a term deposit at the same bank is the price of the foregone liquidity of the term deposit.

Hence if there is an equilibrium between the two securities, then the first must be more expensive.  They will still end up at the same value, thus the first now has worse yield in terms of money; instead it provides liquidity.  As the date of expiry draws near, and the respective prices come closer to each other, the liquidity premium is disappearing because there is now less liquidity service left in the first security.  A fire extinguisher that will stay good for two years is worth about twice as much as one that will have to be thrown out after only one year.

Theories of money

With that, let's look at the three main theories about money.

Perhaps the most widespread theory, expressed in equations like MV=PQ.  In words, this states that, if the quantity of money is doubled, then (eventually) prices will double, and nothing else will change about the economy.  This is called the neutrality of money in the long run.
It is understandable that economists like this theory.  After all, if they assume that money is neutral, then they can separately analyse the real and the monetary economies.  This is also called the classical dichotomy.  In mathematical terms, this is when a multidimensional distribution factorises into the product of two lower-dimensional distributions.
Unfortunately, this theory falls far short of adequacy, and cannot explain many things that happen in reality.
The theory preferred by accountants and security analysts.  Money is a liability of the central bank, and thus a claim on its assets.  Perhaps this claim cannot be exercised at the will of the money-holder, but only by the central bank.  On the other hand, deposits at a commercial bank are claims on the issuing bank's assets—particularly cash, or one's debt towards the bank—that can be exercised at will.
Unfortunately, this theory comes up short, and cannot explain a few phenomena, for instance Bitcoin.
This theory tells a story.  Imagine a state that levies a tax on its people, i.e. it confiscates some of their property, demands that they work for it, etc.  Then, the state announces that it will accept an otherwise worthless token instead of real wealth during taxation.
As long as these tokens that purchase immunity from taxation are in short supply, they will have a market value not much below the value of the taxation they can be substituted for.  However, if there is a glut of them, and everyone has plentiful protection from taxation, they will not give much for yet another token, and thus they will trade at a large discount from face value.
This theory sounds sensible.  Unfortunately, there are many counterexamples to it, where the people started to use something other than the local tax-receivable currency for trade.

Summing the theories

To go back to securities analysis, money is a highly liquid callable couponless perpetual bearer bond.  It has fundamental value as a liability of the central bank or the state, which will accept it at face value in the case of either conversion or tax payment.  And on top of that, it has a significant liquidity premium.

Thus we can essentially add together the backing and quantity theories.  The first determines the fundamental value of money, while the latter its liquidity premium.

This has many interesting consequences.  For one, money is not neutral, and thus the economy cannot be purely decomposed into real and monetary parts even in the long run.  For another, alternative sources of liquidity decrease the premium of money.  Even further, quantitative easing by buying up assets at market price can only decrease the liquidity premium of money, not its fundamental value.  Indeed, if the central bank buys up very liquid securities (such as T-bills), the effect is even more subdued.

In theory, if the central bank bought instruments that are more liquid than its deposits (only available to banks) or cash (bulky and low-yielding), the total amount of liquidity in the economy would decrease, causing money to appreciate, i.e. deflation.  This is consistent with the fact that the central bank would have to pay a market rate above face value for T-bills if they were more liquid than reserves.  However, the central bank would quickly go central bankrupt, if it had to pay more interest on its assets than it could collect on its liabilities.

Liquidity factory

Over the course of history, there have been many institutions whose business has been liquidity creation.  If they could successfully add liquidity to their input, it could be sold on at a profit.

Mints: the average person has a difficult time measuring the weight and especially the purity of irregular bullion.  By turning it into standardized and easily verifiable coins, it becomes easier to do trade.  Thus coins had agio, meaning they traded at a value above that of their intrinsic metal content.  A coin with 10 grams of silver could often buy 11 grams of silver bullion.

Banks: the promise of an average person to pay is not very liquid.  Those who don't know him will only accept it at a very steep discount.  On the other hand, a standardised promise of a large corporation to pay whenever you demand it is much more so.  Hence when the bank and the borrower exchange their promises to pay, those of the bank are more valuable, and the bank can demand more of the borrower's promises, i.e. interest, in return.

Mutual funds: scooping up a wide variety of illiquid shares (or bonds) from smaller businesses as assets and issuing a large volume of uniform shares, the mutual fund can create liquidity.  This is observable only with closed-end funds, because if an open-ended fund were to trade above NPV, investors would expand it; and if it were to go below NPV, investors would redeem shares.

Holding companies: Essentially the same business as mutual funds, except they tend to get involved in running the business. 

The United States: for at least two decades now, America has been running a significant trade deficit.  Many economists have predicted the imminent collapse of the dollar due to its superabundant quantity abroad.  However, these dollars are not superfluous; as the premier international settlement medium, the liquidity premium on dollars in the world trade justifies their existence.  In essence, the US has been exporting valuable liquidity to the rest of the world.

What if I don't need liquidity?

If you only want to invest for your retirement, you have no need for liquidity or market depth.  However, if you hold highly liquid assets, you will end up paying for a feature that you have no use for.  What can you do about that?

The obvious answer is that you can hold illiquid assets.  Real estate, a wide and random-ish selection of small stocks (you know you won't beat the market, thus you rely on the efficient market hypothesis) and similar things are the currently available options.  But what if you want to do something better?

An interesting possibility is the equity deposit.  Similarly to a term deposit at the bank, you forfeit your right to sell the deposited equity, but in exchange you receive higher returns.  This is one side of the transaction—what about the other?  For example, a short seller currently can only borrow shares with a callability clause.  At any point during the deal, the lender might demand back the lent share, and the short seller will have to buy it back—even if at the moment its price is higher than what he sold it at.

However, if that share were lent by an equity deposit, thus it were guaranteed that the lender will not sell it, then it could be lent out without callability.  This would insure the borrower that he will not have to return the share before the contract is up.  As such, short sellers would be willing to pay a higher interest rate for borrowing the share under these terms.

An even more interesting solution would be an all-out liquidity market.  For example, someone who is not satisfied with the liquidity of a particular security could buy a put-at-ask option to go with it.