Bounded growth is when a growth rate is constantly decreasing. The result is that growth approaches a fixed value. In other words, when your growth rate is constantly decreasing it eventually becomes so small that you're essentially sitting still. Technically you may forever approach a bound without ever touching it.
ExampleA startup company with 100 million in sales and a 100% growth rate initially looks like it might achieve exponential growth. However, with each quarter its growth rate falls by 50%. As a result the company never reaches 500 million in revenue as its growth rate becomes extremely small with time: 
Growth Rate  Revenue  100%  100 Million  50%  200 Million  25%  300 Million  12.5%  375 Million  6.25%  421 Million  3.13%  448 Million  1.56%  462 Million  0.78%  469 Million  0.39%  472 Million  Bounded growth is reasonably common as big things tend to have trouble growing, a phenomena known as Diseconomies Of Scale. It is common for a period of exponential growth to be followed by either bounded growth or a period of decline.
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