If I call a business and the receptionists transfers me to an extension. How can I find out either that extension that she sent me to or, even better, how can I find that phone numbers direct line? I have an iPhone, is there an app, or any hack using Skype on my computer... any way at all?

It may not have one. DDI and DIA are expensive telecoms options and the cost may not be justifiable.

Unless the person answering the extension or the receptionist tells you then I doubt there is any way to know.

And as Heinz says there is no reason for a private switchboard extension to have a direct dial in number anyway. And if it does there is no reason why the DDI number has to bear any relationship to either the main switchboard number or the extension number.

Even if that person called you back directly there is also no reason why the CLI sent by their extension has to be the DDI number of their extension. Many modern switchboard systems can be programmed to send any number you like. So all the phones could be programmed to send just the main reception number not the DDI of the individual extension.