It depends what psychological experiment you have in mind. Within a mobile browser for example you can do almost everything today, the big limitation is then the reaction time. If you need reaction times that are more precise then one second, you should use a dedicated app for that. It's also important whether it must be accessible online or offline and which mobile os is supported. Perhaps you refine your question because I doubt there is a product that covers all applications in psychological research on every smartphone.
Agreed with Markus that it really does depend on the nature of the design and the data you are looking to collect. If you search online for "experience sampling" and "ecological momentary assessment" tools you will be able to find reviews from psychologists and visit company websites to see what might be a good match for your needs.