Question: I can imagine that finding $p,q,n$ are easy, but my text book also says the $e$ can be found rapidly, so how does modern RSA find this $e$?

![RSA as a Public Key System](https://i.stack.imgur.com/LBR7F.png)

Answer: 
Clarification
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> I'm wondering whether there exists an encoding/hashing/encryption scheme whereby the original string can always be derived in its entirety given the entire encoded/hashed/encrypted string, and nothing else (no key/password). But also, no portion of the original string can be derived given any portion of the encoded/hashed/encrypted string.
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I am assuming that "no portion of the original string can be derived given any portion of the encoded/hashed/encrypted string" means "no portion of the original string can be derived given anything less than the entire encoded/hashed/encrypted string", otherwise the question would be self-contradictory.

Answer
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It sounds like you are looking for a permutation. A permutation is an invertible transformation on a fixed-size set of blocks. If your input is larger, the/an [All-Or-Nothing Transform](https://en.wikipedia.org/wiki/All-or-nothing_transform) may be useful. The OAEP mentioned by @DannyNiu is an example of an AONT.

For example, many block ciphers are built by interleaving applications of a permutation with the addition of secret key material. The permutation provides diffusion, which ensures that if you modify any part of the output then attempt to invert it, you end up back at a completely different input. 

If you simply strip the key addition portion from a block cipher, it should also do what you're asking. For example, [AES](https://en.wikipedia.org/wiki/Advanced_Encryption_Standard) consists of subBytes, mixColumns, shiftRows, and addRoundKey. If you were to omit the addRoundkey operation, you would be left with a fixed permutation that provides the required avalanche effect and some degree of unpredictability. Another example of a permutation is [keccak-f](https://en.wikipedia.org/wiki/Keccak#The_block_permutation), which does the mixing for the [SHA3 algorithm](http://en.wikipedia.org/wiki/Keccak).

A key-less permutation does not provide encryption
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Note that such a construction with no key is no longer providing *encryption*, as it is not possible to provide confidentiality of the message without some kind of secrecy, which is what the key provides. If anyone who has an input message can compute an output "ciphertext", or anyone who has an output "ciphertext" can invert it to the input message, then clearly confidentiality of the input cannot be achieved.

You tagged this question with "encoding", so perhaps confidentiality is not required in your use case. You would need to establish what you need this construction for and whether or not this is an issue.
Let's take the simplest cipher I can think of which the key is not obvious: [ROT13](https://en.wikipedia.org/wiki/ROT13). You replace a letter with the 13th letter after it, in the alphabet. Because there are 26 letters in the alphabet, you don't need to inverse the algorithm for decryption. Is ROT13 an example of keyless encryption, a very weak one of course? No, because 13 is your key. I don't think any algorithm that can be expressed with a computer can be keyless.

Another bizarre example of which the key is not obvious is a language. Obscure languages have been used as a form of encryption. The most famous example are the [code talkers](https://en.wikipedia.org/wiki/Code_talker) of World War II. Navajo was an unwritten language at that time and the only way to encrypt a message was using your language skills and your mouth. You decrypted a message using your ears, your language skills, and pen and paper. So is this form of encryption keyless? To answer that question, we have to know if this algorithm can be expressed by a computer. There is no way for a computer to translate English to Navajo right now, but there are services that can translate English to Chinese and vice versa. If we trained a language model that could automatically translate English to Navajo using a computer, that model would be your key.

So this answer doesn't ultimatively exclude any form of keyless encryption might exist. I just had a look at some encryption topics not covered by RFC or NIST and I don't think it's possible for keyless encryption to exist.