Human history is considered by many to be path-dependent in the sense that accidental events such as, for example, the death at an early age of major historical figures such as Napoleon or Hitler would have probably have led to very different historical events.
Path-dependency can also apply to economical, social and biological phenomena.
Consider for example the issue of driving on the left or on the right. Technically the two choices are equivalent. The initial choice might therefore have been due to accidental reasons. Once the choice was made, though, it became permanent because of the huge cost involved in modifying it. A similar issue is that of the Rail gauge.
In the computer and software markets, legacy systems indicate path dependence: customers' needs in the present market often include being able to read data, or run programs, from past generations of products. Thus, for instance, a customer may need not merely the best available word processor but rather the best available word processor which can read Microsoft Word files -- a limitation which contributes to proprietary lock-in.
Evolution is considered by some to be path-dependent in the sense that random mutations occurred in the past might have had long term effects on current life forms.
Forecasting the behavior of a path-dependent system might be harder as the system is less likely to converge as random events can disrupt its evolution.