Typical examples are Asian options, which use the average price over a period; lookback options, which use the highest or lowest price; and barrier options, where the right changes depending on whether a set level is touched. Even with the same price at expiry, a different path before expiry gives a different payoff.
Many types can be valued without the whole path, using only the average or the highest and lowest prices up to the valuation date together with the current price.