What base is this number in?

Python, 27 22 bytes

lambda s:(max(s)-8)%39

This requires the input to be a bytestring (Python 3) or a bytearray (Python 2 and 3).

Thanks to @AleksiTorhamo for golfing off 5 bytes!

Test it on Ideone.

How it works

We begin by taking the maximum of the string. This the code points of letters are higher than the code points of digits, this maximal character is also the maximal base 36 digit.

The code point of '0' – '9' are 48 – 57, so we must subtract 48 from their code points to compute the corresponding digits, or 47 to compute the lowest possible base. Similarly, the code points of the letters 'a' – 'z' are 97 – 122. Since 'a' represents the digit with value 10, we must subtract 87 from their code points to compute the corresponding digits, or 86 to compute the lowest possible base. One way to achieve this is as follows.

The difference between 97 and 58 (':', the character after '9') is 39, so taking the code points modulo 39 can achieve the subtraction. Since 48 % 39 = 9, and the desired result for the character '0' is 1, we first subtract 8 before taking the result modulo 39. Subtracting first is necessary since otherwise 'u' % 39 = 117 % 39 = 0.

c    n    n-8    (n-8)%39
0    48    40     1
1    49    41     2
2    50    42     3
3    51    43     4
4    52    44     5
5    53    45     6
6    54    46     7
7    55    47     8
8    56    48     9
9    57    49    10
a    97    89    11
b    98    90    12
c    99    91    13
d   100    92    14
e   101    93    15
f   102    94    16
g   103    95    17
h   104    96    18
i   105    97    19
j   106    98    20
k   107    99    21
l   108   100    22
m   109   101    23
n   110   102    24
o   111   103    25
p   112   104    26
q   113   105    27
r   114   106    28
s   115   107    29
t   116   108    30
u   117   109    31
v   118   110    32
w   119   111    33
x   120   112    34
y   121   113    35
z   122   114    36

Python, 25 bytes

lambda x:int(max(x),36)+1

Defines a lambda that takes the string x. Finds the largest digit in the string (sorted with letters above digits, by python's default), and converts to base 36. Adds 1, because 8 is not in base 8.

Jelly, 4 bytes

ṀØBi

Requires uppercase. Try it online! or verify all test cases.

How it works

ṀØBi  Main link. Arguments: s (string)

Ṁ     Yield the maximum of s.
 ØB   Yield "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz".
   i  Find the 1-based index of the maximum in that string.