Deciphering text encrypted with a changing cipher I know there are many ways to crack basic ciphers were each letter is mapped to some other letter, but what ways are there to decode something that was encrypted using a cipher that changed after every letter, in a way that is based on the letter just coded?

For example, imagine using a Caesar cipher where the number of letters the alphabet was shifted over changes by $n$ after every letter coded, where $n$ is the position in the alphabet of the letter just encoded.

That would seem to null the efficacy of observing letter frequency, double letter patters, common short words, etc.  How would a code like this be cracked?

Sorry, I might have been unclear.  The Caesar cipher was just an example, but say that the positions shift over $f(n)$ letters instead of just $n$, how would $f(n)$ be determined?
It's insecure because the algorithm you described has essentially no key. You just have to try 25 different offsets for the first character, then run the algorithm and see if plaintext comes out.

Have a look at Kerckhoff's Principle, and try to understand why there needs to be a secret key in addition to an algorithm.