Your plans seem to be rather fluid, to put it mildly. First you ask what there's to bo in Peru, and then go on to say you'd like to include Easter Island (Chile, though way far from the mainland) and French Guyana.

I doubt open-jaw is a term you'll see much in booking flights. You're talking about some options that cover very, very big distances and would likely cost you a lot of money. If you're on a tight budget, then trying to get to far-flung places served by few carriers, such as Easter Island and Cayenne, may well not be feasible. Your question is so open-ended right now, that I'd suggest getting a guide book to South America (e.g., Lonely Planet, Rough Guide, which have advice for "budget" travelers), and play around with different itineraries as far as travel is concerned.

I've not been to Easter Island but we did look into it, and found flights, and accommodations, to be very expensive. Maybe not enough to do to justify the high prices. Have you actually looked into travel in French Guyana or just thought it sounded like an exotic, unusual place to visit? I've been to Cusco and Machu Picchu and enjoyed both. I'm in Brazil, which, of course, is a huge country with lots to do, but which is not cheap (including flighs).

I'd start with the guide book and look at flight prices for your specific dates. And consider sticking to one region, such as Peru and Ecuador, or even Chile/Argentina (Patagonia is spectacular, if you go in their summer). Whether there'd be enough to occupy you for 3 weeks depends on your interests.