Using Data Compression on the output of Data Compression
When we compress something the output is smaller than input (that is the purpose of compression, otherwise we do not use it or cope with bigger file). This can be achieved by various methods including (but not limited to) matching of substrings (dictionary compression) or frequency of characters (entropy compression, like Huffman encoding) or arithmetic coding (like MTF, Elias etc.).
Some methods like (Bijective) Burrows Wheeler Transform does not compress, just change the order of characters so other methods might benefit from local similarity of characters, so it is used in conjunction with other techniques.

If we try to apply dictionary method or entropy method twice (or more times) there are no benefits at all, we already exploited the redundancy of data. Using methods with chunks of data twice will give us the issue of compressing the packed data and the header (dictionary, frame, characters etc.), so it is harder task than one we started with and the most benefitial would be changing frame size or make it addaptive (but not applying the same method more times).

Encryption is not concerned about file size - it secures data with key so that any person obtaining the file without the key will have very hard time opening it. If the quality of encryption is useable the resulting file has no obvious dependencies or visible redundancy. Encrypted files are not compressible. Now encrypting chunks several times will not add any security and will not compress file.

Please keep in mind that methods like Cezar, Rot13 and xor of key (cyclic) with data are not encryption methods, and even if they were, cyclic shift of characters (mod 256) will not change compression.

About limits - when we compress something then we have to encode header, dictionary etc., so our output file is transformed input plus additional information about it, which is commonly short and not compressible, so every time we put output of compression into input of another one we have a bit more to deal with and some redundancies were already used, this is one limit.
The second limit is that we cannot compress anything in the mean sense, taking all possible inputs of given size, compressing and calculating the mean shows that we in fact ended with more data than we started with. At the best (unlikely) case we end up with the same amount of data. This comes from the fact that contraction of data with better result is not unique anymore so we cannot decompress it uniquely to original.