Unexpected symbol `public`

Use code tags!

Also look at this line right here:

void StartGame();

Anything look weird? Like maybe a misplaced semi-colon?

From there the rest of the body of ‘StartGame’ is not wrapped correctly, making ‘NextGuest’ malformed.

You may notice if you had it formatted with code tags:

using UnityEngine;
using System.Collections;

public class NumberWizard : MonoBehaviour {

    int max = 1000 ;
    int min = 1 ;
    int guess = 500;

    // Use this for initialization
    void Start () {
        StartGame ();

    }
    void StartGame();
        max = max + 1;

        print ("Welcome to number wizard!");
        print ("Pick a number in you head but don't tell me");


        print ("The highest number you can chose is " + max );
        print ("The lowest number you can chose is " + min );

        print ("Is you number higher or lower than " + guess);
        print ("Up key for higher, down key for lower, enter for the same");

    void NextGuest () {
        guess = (max + min) / 2;
        print ("Is it higher or lower than " + guess);
    }

    // Update is called once per frame
    void Update () {
        if (Input.GetKeyDown(KeyCode.UpArrow)) {
            min = guess;
            NextGuest ();
        }else if (Input.GetKeyDown(KeyCode.DownArrow)) {
            print ("down arrow pressed");
            max = guess;
            NextGuest (); 
        }else if (Input.GetKeyDown(KeyCode.Return)) {
            print ("I won!!!");

        }
    }
}

Note that usually errors/exceptions will point directly at the offending line of code.