Minimum base to count to a string

Python 2, 74 64 56 55 53 bytes

def f(x):a=ord(max(x))-54;a+=7*(a<8);print int(x,a),a

Call the function (i.e. f("HELLOWORLD")) and outputs to stdout (809608041709942 33)

This is case-sensitive, requiring upper case letters, and crashes for invalid input (e.g. "$^$%!(&£%)())

This code gets the max letter in the string (z>x>y>...>b>a>9>8>...>2>1>0) and gets the largest base needed to represent it.

Change the 54 into a 76 to make it work for lowercase (but no longer upper case).

APL (24)

{(B⊥D-1),B←⌈/D←⍵⍳⍨⎕D,⎕A}

This is a function that takes a string and returns two values, the value and the base.

      {(B⊥D-1),B←⌈/D←⍵⍳⍨⎕D,⎕A} 'DEADBEEF'
3735928559 16

Explanation:

• ⎕D,⎕A: digits followed by letters
• D←⍵⍳⍨: store in D the 1-based index of each character of ⍵ in the string of digits followed by letters
• B←⌈/D: store in B the highest value in D. (which is the base to use)
• (B⊥D-1): subtract 1 from each value in D (making them the values of the digits), and decode them as a base-B number.

Pure Bash, 48

for((b=37;$[--b]#$1;)){ :;}
echo $[$[++b]#$1],$b

Output

$ ./minbase.sh HELLOWORLD
./minbase.sh: line 1: ((: 32#HELLOWORLD: value too great for base (error token is "32#HELLOWORLD")
809608041709942,33
$ ./minbase.sh HELLOWORLD 2>/dev/null
809608041709942,33
$ ./minbase.sh 1 2>/dev/null
1,2
$ 

I'm taking advantage of this meta answer - specifically that warnings output to STDERR may be ignored.