Is there a cipher that avoids double bigrams? Are there any known ciphers that whose ciphertext (given any plaintext) avoids bigrams of the form "aa, bb, cc, etc."?

What about known ciphers that make bigrams of form "aa, bb, cc, etc." rare compared to other bigrams?

I understand that it is easy to construct a cipher that does this, but I am wondering if there are any in the literature that have this property.
The Playfair cipher would partially avoid this in that it does not encipher two letters that are the same. So the letter pair "LL" would be enciphered as "LX" "L_" (or sometimes just "LX"), producing different ciphertext letters for each pair.  It would allow ciphertext letter repeats of the form "AB BC" where "AB" is one pair of letters that are enciphered together, and "BC" is the next pair of letters that are enciphered together.  Indeed, this would occur when enciphering "at ta" (such as in "attack").

Looking at a list of classical ciphers, I don't see any others that would even avoid double letters to that extent.

You may want to look here which discusses the frequency of double letters in bigram frequencies.  In particular, "LL" is the most common double letter bigram in English, and  occurs 0.577% of the time, and only "LL", "SS", "EE", "OO", and "TT" (and almost "FF") beat the random bigram frequency of 0.148%.  I'm not sure if this counts repeats of the form "at the" or not.  So if the bigram frequency depends on the plaintext, double letters bigrams would generally be less common than other bigrams.

Another factor that probably comes into play is that "aa" is more noticeable than "ax", even if the frequencies are the same.