Asian options: strike is average of underlying over tenor. Underlying is stochastic.
Options with kock-ins/knock-outs: Underlying is stochastic and may cross the kock threshold as it evolves. Option value depends on this cross or lack thereof (boolean).
Options on Options, too.
Motivations for Asian options you can google. Kock-ins and knock-outs lower the cost of an option so that a buyer who needs them for hedging purposes can more cheaply acquire downside protection. Generally applies to Currency/Interest Rate hedging.
Options on options are more of a theoretical construct, although I'm sure they exist in practice. Generally useful for valuing complex projects with optionality when standard DCF approaches would not fully capture the stochasticity of the project's value. Dixit has a good chapter on this, I believe.