How can the stock market keep growing indefinitely?

The total value of assets can exceed the total amount of cash or money in accounts.

For example, suppose we live in an economy with exactly as much money as assets (say, 100 trillion). I then produce, at my home, a masterpiece of art from common materials, akin to the Mona Lisa. Appraisers and others assess it at 1 trillion dollars. There does not need to be a corresponding increase in paper money or bank accounts, but the total value of assets is now increased beyond 100 trillion.

As long as each stock, individually, can be purchased with the amount of money in the system, it's not an issue that we can't buy all of them at once.

As mentioned in the other answer, the amount of money does not have to correspond to the total value of all assets in the economy. However, there is some correspondence that is not mentioned in the other answer so I will focus on that.

First, there should always be enough money in the economy so people can carry all the transactions they want. If that is not true the money market would not be in equilibrium. This can be formalized using the equation of exchange (see Mankiw, Macroeconomics 7th ed pp 86):

$$MV=P_TT$$

where $M$ is the stock of money, $V$ is the velocity of money (how many times is on average one bill used), $P_T$ are the prices and $T$ is the amount of transactions (which for the purposes of this answer can include asset transactions as well although normally this sort of model would be used only for goods and services).

We can learn several things from this formalization.

First, as the previous answer pointed out amount of money $M$ does not need to be equal to the value of assets, since the equation is $MV=P_TT$ not $M=P_AA$ (where $A$ would be assets). So the velocity of money matters as well. If you buy from me 1 stock of Apple for ${$}100$ I can use the same ${$}100$ dollar bill to buy other assets - so even though there is only single ${$}100$ dollar bill multiple things can be purchased with it if it circulates fast enough. Moreover, because not all assets $(A)$ are always transacted $(T)$ there does not need to be correspondence between value of all assets and money as well.

Second the formula $MV=P_TT$ can be rearranged by decomposing the prices into aggregate price level $P$ and relative prices $P_r$ of the 'things' that are being sold on the market where the following equality would hold $P_TT=P P_r Q$ where $P_r Q$ would be the real value (i.e. inflation adjusted value) of $Q$'s that are being sold (which we could also be in principle some asset). Lets also denote the real value of things being sold as $T_r$. So now the model would look like:

$$MV = PT_r$$

The equation above also shows us another thing. Even when $M$ and $V$ are fixed the real value of transactions $T_r$ can grow because prices adjust.

Ultimately what matters are real returns not nominal returns. When someone says that stock market returns beat inflation by $7%$ that means that they real value increased by $7%$ not that their nominal value is by $7%$ higher. For example, imagine that economy is experiencing deflation (negative inflation of $10%$ and stock increases in value in real terms by $7%$ per year. In such such situation if you invest ${$}100$ dollars for one year in nominal terms at the end of next year your stock will be worth only ${$}97$. However, real value of your asset (correcting for inflation or in this case deflation) is ${$}107$ because deflation means everything in economy is getting cheaper and so you can purchase much more with ${$}97$ than with ${$}100$ year before.

The real value of asset such as stock depends, very broadly speaking, on peoples' valuation of the underlaying companies. Constant improvement in technology and productivity on average leads to companies being more valuable on average over long time periods (of course, technological progress destroys some companies but it also allows other to flourish or completely new industries to rise up and we are talking about whole market here). Consequently, in real terms the only limit for how much stock market can grow are technological & resource constraints avaiable to humanity (with emphasis on the technology).

In addition, it is worth noting there is no limit on how many money there can be in the world. Nowadays most of money is not even printed it is just created at a keystroke virtually. In principle if central banks would be willing to they could always find a way to expand money supply to any size they desire, so in theory $M$ can be any non-negative real number you can think of.

Hence in conclusion, the premise in your question ignores the velocity of money and the fact that not all assets are always transacted so there does not need to be enough money to purchase them all. And secondly even if velocity of money and and money supply is fixed the prices can still adjust in a way that allows for the market to grow in real terms.

Social customs in commerce and government impose an upward bias so investors tend to average up the last sale price of equity shares on secondary markets over time. This social process might come to an end via natural disaster or social disruption where compound increase in equity prices could be a feedback to the crash of market prices such as via overshoot and collapse pattern.

Technically cash positions in brokerage accounts are invested in the money market mutual fund (MMMF) industry. Assuming both buyer and seller hold cash positions in MMMF shares then whenever shares of stock trade on the secondary market these transactions do not change levels of MMMF shares or shares of stock. The buyer and seller swap title to MMMF shares and shares of stock.

In aggregate the "risk off" trade means many investors prefer to own MMMF shares and try to sell shares of stock. This does not increase MMMF levels. Instead prices for shares of stock will average down rapidly at the last sale and the MMMF industry has no opportunity to increase levels of funds via this mechanism. If investors prefer short term Treasuries and insured bank deposits to MMMF shares that threaten to "break the buck" then even levels of MMMF shares will be reduced by the "risk off" sentiment.

In aggregate the "risk on" trade means many investors prefer to own shares of stock and try to minimize holding of MMMF shares. This does not decrease MMMF levels. Instead prices for shares of stock will average up at the last sale and the MMMF industry has no opportunity or desire to decrease levels of funds via this mechanism.

During the so-called Global Financial Crisis in late 2008 the effort to take risk off in financial markets could not be satisfied by balance sheet adjustments of banks or other financial intermediaries because levels of money and MMMF funds are a consequence of deals made on the financial intermediary balance sheets. Market forces were causing many banks and other financial intermediaries to sell assets and reduce (unwind) their balance sheets. The U.S. Treasury and Federal Reserve System expanded their balance sheets to provide some liquidity for the "risk off" demands in the global financial markets.

Aren't you assuming the stock market keeps growing indefinitely? This is wrong.

I'm reading Irrational Exuberance (2015) by Robert Shiller, Nobel Economics Laureate in 2013, Sterling Professor of Economics at Yale University. p 74

      Where did people get the idea that, if there is ever a stock market crash, the market is sure to rise to past levels within a couple of years or so? History certainly does not suggest this. There are many examples of markets that have done poorly over long intervals of time. To pick just one from recent memory, the Nikkei index in Japan is still selling at less than half its peak value in 1989. Other examples are the periods after the 1929 and 1966 stock market peaks discussed in Chapter 1. However, during a booming market, these examples of persistent bad performance in the stock market are not prominent in the public mind.

The other answers focus on the stock market aspect, I'll add somehting else. You say in your question that there is a fixed amount of money and that the central banks control this, but most money is created by commercial banks. You can read more at https://en.wikipedia.org/wiki/Money_supply. Here you will see several statistics that show that cash and central bank money is just a fraction of all money and the total amount has constantly been increasing. So money supply is not a limiting factor to the stock market. Hope this helps.

A stock represents equity in a company, and as such, is a claim on the future earnings of the company. The more the company is anticipated to earn in the future, the more the stock is worth. So all the stock in the stock market represents claims on all the future profits of all companies in the market. It's a promise of future money, with a somewhat tenuous relationship to the actual flow of money through the rest of the world economy.

Stock prices can rise without limit--or rather are only limited by peoples perceptions of how much money they might someday be able to lay claim to. Valuations are sometimes...unrealistic. And when someone realizes that, and tries to get out...prices fall. Failing stock prices then induce others to sell, and you get a panic, and the stock market crashes. The everyone feels less rich and spends less money. So the economy contracts.

Go read the 'Economix' graphic novel for a better understanding of the economy.