Geometry Lesson 
Is it me or does the teacher in the picture not look happy?
That's odd because we're going to talk about compound growth, which is a seemingly pleasant financial principle. Compounding is supposed to be a boon to investors and by the way, also mankind's greatest invention.
The idea is if you invest for long periods of time you'll get exponential returns. Exponential is a fancy word for bigger and bigger.
Maybe the teacher is frowning because this principle, while true on paper, is often exaggerated as to how it works in the real world...
What is Compounding?
Compounding is the process whereby reinvested returns result in the investment earnings increasing exponentially over time. A simpler definition might be: "when you earn interest on your interest." Over time, you earn not just on your original investment, but also on the earnings themselves.
This is the basis for the idea that if someone forgot about $5 in a bank account during the Civil War, today it would have grown to millions due to compound interest.
The chart below shows how geometric growth looks. Notice the line is not straight, but curved, and the curve becomes increasingly pronounced. This is called the "hockey stick."
Perhaps geometric growth worked for Mark Zuckerberg  but can it work for the rest of us?
Mark Zuckerberg opens the Facebook f8 conference with user growth numbers alongside major product launches. 
The Amount of Capital Affects the Amount of Return
The rich do get richer. But not necessarily due to compounding. One reason is the more money invested, the more earned on the same positive percentage return.
If $10,000 invested earns 2%, that's $200. But if instead of $10,000, it's $1,000,000, the earnings would be $20,000. The percentage return stayed the same, but a greater gain resulted.
The simplest way to increase earnings is to have more invested. I find it interesting we so often focus on the return, when the easiest way to boost earnings is to increase the amount of capital put to work. Of course, having additional capital may not be an option.
Compounding and the Rule of 72
The “Rule of 72” is a rule of thumb. Divide a rate of return into the number 72, and this will tell how long it would take to double. For example, a 10% annual return means a double in 7 years (72/10 = 7). A 2% return means it would take 36 years (72/2=36).
If you start inputting returns, you'll notice a small increase in the rate of return produces a big difference in the final product.
Consider someone, age 50, investing $10,000. This table shows what the Rule of 72 would imply with a return of 10% per year:
Hypothetical Investor: Age 50
Rate of Return: 10%
Time to Double: 7 Years
 
Age

Investment Sum

50

$10,000

57

$20,000

64

$40,000

71

$80,000

78

$160,000

85

$320,000

Now compare the result if our only earned 2% per year. The time to double would be 31 years.
Hypothetical Investor: Age 50
Rate of Return: 4%
Time to Double: 18 Years
 
Age

Investment Sum

50

$10,000

68

$20,000

86

$40,000

After looking at these figures, it's easy to conclude compounding is a great deal. Investing in riskier assets would seem to be the way to go.
And there is mathematical truth to the idea. Investing for a longer period and earning a higher return will result in an exponentially higher sum.
Charts such as the above have shaped much of our investing philosophy. The key is to put the money in for a long period of time and to try to find investments which pay a higher incremental return.
Hmm...
Could Compounding be Overrated?
The problem is the concept lends itself to oversimplification. I believe many attempt to benefit from compounding and wind up losing.
While it's great when it occurs, I question how often this will be. Indeed, some of the arguments break down upon inspection.
You may have heard Albert Einstein once said compound interest is modern man's “greatest invention,” or something to this effect. In my line of work, I have heard it a thousand times. However upon researching the matter, I can find no evidence he said this. It seems to be a statement that has been attributed to him to increase the idea that compounding is a magic elixir. It's modern day folklore.
Factors that Mitigate Compounding
First, compounding becomes a major factor only after the money is left untouched for an very long period of time. How many investors can afford to leave money untouched in an investment for 20, 30, or 50 years? A few, but not many. And probably not retirees. Most retirees need to draw income from their investments.
Second, to benefit from compounding an investor needs to earn a high return for a long period of time. This is difficult to do. It is hard enough to earn a high rate of return for a short period of time. Remember the arithmetic of loss? Losses disproportionately affect your rate of return.
Third, your investments are not the only thing compounding. Inflation, has averaged an annual rate of about 3% over the past 100 years. So while your investment may be growing at a
compounded rate, so is inflation. Inflation diminishes the value of your principal over time, much as compounding increases it.
Rip Van Winkle's Unhappy Discovery
The investor is elated when he is told the value of his account has grown to $100,000. It seems a dream come true... everyone wishes they had a hundred years to be invested and to let their money compound. Then a voice comes on the phone and says the cost of the phone call is $500 dollars.
The idea is that while the savings compounded, so did inflation, and much of the effect was mitigated by the inflation.
Your interest or earnings may compound, but so does inflation!
Conclusion
Compounding may indeed be a positive factor if you are a long term investor. With a long term horizon, and if you are fortunate enough to earn a consistent positive return, the effect can be significant.
But be careful facile arguments. Don't let the hope of compounding draw you into unsuitable levels of risk.