I have read the importance of saving early and the power of compound interest many many times. However I am a Muslim and cannot participate in anything involving interest. I also learned that keeping money in terms of just cash is the worst form since there is inflation etc. I would like to know what can I do to become rich. Please suggest. Thanks.
There are many Shariah compliant investments, so that could direct your resulting searches. Shariah compliance is a very strict interpretation of Islam and for investing offers strict guidelines in what to invest in and excludes investments in companies that engage in certain businesses such as gambling, tobacco, pork and trading of gold and silver on a deferred basis (and more). Many multinational financial service companies such as the Standard & Poors (S&P) offer Shariah Compliant funds and indices, as such, it makes it easier to invest in a variety of different assets through them. You can also look at their fund's constituents and invest in those assets directly.
Secondly, going back to your original question about a compound interest equivalent, you can look at the products offered by Shariah Compliant banks.
Now, if it is really important for you to adhere to the strictest interpretations of your faith, you should know that most Islamic Banks have interest bearing assets within them and that they disguise that fact. The global financial system is based on interest bearing instruments such as bonds, and Islamic banks are large holders and issuers of those instruments, and all of their consumer products are also based on the interest rates of them. Even convoluted alternatives such as Islamic mortgages, where they are advertised as non-interest bearing equivalents, many times are also the interest bearing version. Unfortunately, these lies are enough for the banks to continue to get business from their target audiences, but outside of Islam this is a very standard and stable business practice.
The point is that you should look very carefully at the alternatives you find.
My understanding of Muslim finance is that you may not lend money at interest, including investing in in things that pay interest. However you may still make investments: it just has to be in places where you get a share of profit, rather than a fixed rate of return. You would be better asking the Muslim community specifically for more details.
The benefits of compound interest apply, more or less, to other non-fixed-interest investments. If you invest $1000 in a business and get a 10% rate of return, you have $1100 to invest in your next venture, which means it will be more profitable and so on. That's why the growth happens, not specifically because it is interest. Stocks do not pay interest, and the 'magic' applies to them too. The fact that you might lose as well as win complicates things, but doesn't change the principle.
I cannot tell you what is or is not allowed under Islamic law. What I can tell you is that when most investors talk about the "power of compound interest," they are not actually necessarily talking about interest!
The idea of the magic of compound interest is that when you receive an interest payment on your investment, you now have a larger investment, earning more interest. Your investment grows exponentially. This doesn't just apply to interest payments, however, but can apply to any type of investment where the profits of the investment cause the investment to get larger.
For example, if you invest in a company's stock, and the value of the stock goes up 10% in a year, after that year your investment is worth more than it was at the beginning. If it goes up another 10% the following year, you have gained more money in the second year than you did in the first. Your gains are compounding, even though interest payments are not involved at all. The same is true if you reinvest dividends or if you use business profit to expand your business, for example. The term "power of compound interest" is so named for historical reasons, but really applies to any type of investment where the investment itself is growing.
Invest in growth stocks which do not pay any dividends (Note that some part of the dividends issued by a corporation might be from interest received by the company and passed on to you as a dividend);
Buy a house from a bank that practices Islamic Banking. See this question which you yourself answered a few weeks ago to understand how this works.
Because so many businesses make some money through some form of compound interest, like a business that saves its earnings in a business account that pays interest, it heavily depends on how strict you interpret this law. Some Muslims I know interpret it to mean directly and indirectly, while for some it's just direct interest earned.
What I would suggest is either a direct investment in agriculture or a share in agriculture, where you are directly paid from your share in the investment and not through money that comes from a bank account earning interest. If you do a direct investment in agriculture, like owning livestock, you will be paid money in the form of food, which compounds through reproduction and can sell the offspring to others and collect the money. Year to date, agriculture is crushing the S&P 500 and many places around the world are facing shortages in food, like sugar and corn.
If you don't have enough money for a direct investment, you can try the share route where you own a share of a direct investment. Rather than go through stock exchanges, where many of these companies make money indirectly through interest also, you can negotiate directly with farmers, ranchers, livestock owners, etc. Some of these individuals are looking to diversify their money, so they may be willing to let you own a fraction of what they produce and pay you directly.
All of this comes with risk, of course. Livestock and plants die for a variety of reasons, but none of it will be interest from lending whether to individuals or through a bank. In addition, if we experience very high inflation in the future, livestock and plants do very well in this environment.
I am not sure if these are available today in your country: but supposedly, back when Catholic countries similarly forbade usury, sinecures were invented to circumvent religious restrictions on finance. Meaning literally 'without care', sinecures were formally prestigious salary-paying jobs with few responsibilities. They were bought by the wealthy from the Church or State. The salaries for sinecures, accumulated over time, exceeded the initial purchase price. As such, some moderns consider sinecures usury in all but name.
It depends whether you want to be technically compliant with the letter of the law or compliant with the underlying meaning. For instance, in some countries you can find shell companies that do nothing but deal in fixed income instruments (those that you want to avoid) and dividend stocks (those that you might or might not be allowed to use). You can buy stock of that shell company, which does not hand out dividends itself. Thereby, you transform interest and dividends into capital gains.
These shell companies exist for fiscal reasons, the more risky capital gains are often less taxed than interest or dividends. This might technically solve your problem, but not really change anything in the underlying reality.
P.S. Don't worry too much about missing compounding interest. The rates are incredibly low right now.