* means 0 or more but re.search would return only the first match. Here the first match is an empty string. So you get an empty string as output.
Change * to + to get the desired output.
>>> re.search(r'i*','biiiiiiiiiiiiiig').group()
''
>>> re.search(r'i+','biiiiiiiiiiiiiig').group()
'iiiiiiiiiiiiii'
Consider this example.
>>> re.search(r'i*','biiiiiiiiiiiiiig').group()
''
>>> re.search(r'i*','iiiiiiiiiiiiiig').group()
'iiiiiiiiiiiiii'
Here i* returns iiiiiiiiiiiiii because at first , the regex engine tries to match zero or more times of i. Once it finds i at the very first, it matches greedily all the i's like in the second example, so you get iiiiiiii as output and if the i is not at the first (consider this biiiiiiig string), i* pattern would match all the empty string before the every non-match, in our case it matches all the empty strings that exists before b and g. Because re.search returns only the first match, you should get an empty string because of the non-match b at the first.
Why i got three empty strings as output in the below example?
>>> re.findall(r'i*','biiiiiiiiiiiiiig')
['', 'iiiiiiiiiiiiii', '', '']
As i explained earlier, for every non-match you should get an empty string as match. Let me explain. Regex engine parses the input from left to right.
First empty string as output is because the pattern i* won't match the character b but it matches the empty string which exists before the b.
Now the engine moves to the next character that is i which would be matched by our pattern i*, so it greedily matches the following i's . So you get iiiiiiiiiiiiii as the second.
After matching all the i's, it moves to the next character that is g which isn't matched by our pattern i* . So i* matches the empty string before the non-match g. That's the reason for the third empty string.
Now our pattern i* matches the empty string which exists before the end of the line. That's the reason for fourth empty string.