I have a project in mind that uses Raspberry Pi to develop a software. The final product will be modification of the Raspberry Pi circuit (get rid of any components that the project doesn't use) and add in some other circuits of mine. For the software parts, it will be running Raspbian OS with my application software.
Is it legal to do that? I don't mind giving back if I can make a profit.
The answer to the question if something is legal or not, depends on the interpretation of laws in a particular jurisdiction. This answer should not be treated as a legal advice. It is based on published information by the relevant party.
Raspberry Pi Foundation made a clear statement that answers your question in the Starting a business with a Raspberry Pi article:
If [ ] you’re making a product which requires a Raspberry Pi to run, we don’t ask you to buy special permission or licences from us to use it. All we ask is that you include the words “Powered by Raspberry Pi” somewhere on your packaging. If your business is successful, we’d be very grateful if you could consider donating a small portion of your profits to the Raspberry Pi Foundation – but that’s all, and if you choose not to do that, that’s fine too.
Regarding the use of the "Raspberry Pi" trademark / logo the guidelines are published on the Trademark Rules And Brand Guidelines page.
You can get Element14 to make custom Raspberry Pi's for you:
http://www.technewsworld.com/story/Element-14-to-Bake-Custom-Raspberry-Pis-82670.html
Unfortunately - For whatever reason (most probably exclusivity agreements with Element14) the 2835/2836 is not available for retail purchase. So purchasing that chip from any other channel except Element14 is impossible for now. These agreements tend to run out after 5 to 10 years and cannot be renegotiated as they then may break unfair trade laws in the EU and US.
If you are making your own circuit based on the Raspberry Pi then yes it is legal. Because the PCB schematics is open source (in a limited form) The problem is... you will not get BCM chips because the manufacturer (Element14) most likely has exclusivity on that chip. I tried and it's impossible (the only way they will be interested is if you buy 500k chips at once.. good luck because Google wanted that in the Chromecast and couldn't even get it)(and don't buy Alibaba based chips because they are fakes or factory back door chips that did not pass QA)
If you are going to be buying Pi boards and modifying them then you still need to credit Raspberry Pi foundation even if you remove the logo's and any traces of anything in any way because those boards have specific licenses on them that are provided by the foundation.
You can use Raspbian without permission from foundation because it is based on Debian. You need to read their license page and credit in the appropriate places. You can use any operating system that runs on ARMv7 as long as you adhere to their license policies. Raspbian just includes some "nice" stuff for education, etc,etc but the packages are available across Linux and BSD now. Plus Windows packages emerging slowly too.
--Edit March 2017
As of this month, the Raspberry Pi is the 3rd best selling computer of all time, selling more than 12.5 million boards in the last five years. That is pretty decent since PC and Mac have been selling computers for 30+ years and take 1st and 2nd place.
I personally don't see Element14 letting go of this gold mine easily.
