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All your samples explain easily if you consider that two adjacent string 
literals are joined together. (You seem to consider that "double quote" 
is a way to insert a quote character. It's not; Python is not Pascal or 
SQL)
Your first two examples become:
a='a''a' two adjacent strings == 'aa'
a='a''''a' three adjacent strings ('a' + '' + 'a') == 'aa'
The third is an error:
a='a''''''a' one string ('a') followed by the beginning of a "triple 
quoted string" (''') which content starts with the characters (''a') but 
does not have a matching (''') to finish the string, hence the Syntax 
error.
... and so on. Please have a look at http://docs.python.org/reference/lexical_analysis.html#string-literals