Number that can eat itself
JavaScript (ES6),  52 51  50 bytes
Saved 1 byte thanks to @tsh
Takes input as a string. Returns a Boolean value.

f=n=&gt;n&gt;[n%10]?f(-(-n[0]-n)%10+n.slice(1,-1)):!n[1]

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Commented
f = n =&gt;                 // f = recursive function taking n (a string)
  n &gt; [n % 10]           // The last digit is isolated with n % 10 and turned into a
                         // singleton array, which is eventually coerced to a string
                         // when the comparison occurs.
                         // So we do a lexicographical comparison between n and its
                         // last digit (e.g. '231'&gt;'1' and '202'&gt;'2', but '213'&lt;'3').
  ?                      // If the above result is true:
    f(                   //   do a recursive call:
      -(-n[0] - n) % 10  //     We compute (int(first_digit) + int(n)) mod 10. There's no
                         //     need to isolate the last digit since we do a mod 10 anyway.
      + n.slice(1, -1)   //     We add the middle part, as a string. It may be empty.
    )                    //   end of recursive call
  :                      // else:
    !n[1]                //   return true if n has only 1 digit, or false otherwise