Without any imports, you can use a regular dictionary.
def checkio(text: str) -> str:
lowercase_input = text.lower()
x = {}
for i in lowercase_input:
x[i] = x.get(i, 0) + 1
return max(x.items(), key=lambda x: x[1])
key, value = checkio('asdafsasaaaa') # ('a', 7)
Explanation
dict.get has an optional second parameter, which we set to 0. This means if the key does not exist in the dictionary x.get(i, 0) will return 0. max has an optional key parameter which accepts an anonymous (lambda) function. Since dict.items() returns an iterable of (key, value) pairs, we can calculate the maximum by looking at the value component (1st index).
Performance
This is inefficient versus Counter + most_common as shown by @abc. Specifically, most_common uses heapq to reduce time complexity. See also Python collections.Counter: most_common complexity.