There really is no value in double or triple encrypting data in the manner you describe. First, as a non-cryptographer, you cannot know the level to which you may compromise the crypto process. You absolutely can do that when combining algorithms. A good example of this is 3DES, which is really 3 cipher rounds of DES with the 144 bit key being three 48 bit keys, is more secure doing ENC->DEC->ENC than doing ENC->ENC->ENC. How do I know this? Some crypto guy told me. How did he know? He understood the cryptanalysis.
Information protection does not layer up the same way physical protection does (fences, cameras, locked doors, armed guards, key swipes and passcode, etc.). Some solutions might work like physical protection, but us normal humans can't really tell.
Also, if it did work really well, why aren't cryptographers advocating for the types of layered crypto you describe? I suspect the answer is that the value add (if there is value) is not worth the effort or complexity.
Also, packing keys along with the cryptext (if I understood your proposal) sounds like one need only break the exterior cipher in order to peel back every layer of the onion and get the inside text.
You've inadvertently hit upon the most significant problem with encrypting data at rest (stored data crypto): what to do with the encryption key? A large crypto key makes the data safe. You can't memorize it and you certainly can't memorize a 2048 RSA private key sequence. Where do you put it? How do you protect? You could encrypt it, but then that becomes a recursive problem - what to do with the new key? Ultimately, some human needs to open a wallet or a key store of some type, which means that human has to memorize a long sequence of characters containing enough entropy (true randomness) to be worthy of the data and/or keys it's protecting. That's why people are enamored with USB fobs. Of course, that's a physical object, which is a PITA to carry around and, if lost, often results in loss of all keys and plaintext. In the end, the human may write down the passwords and put them behind a locked door or drawer. So, back to physical security...