Time Complexity for Array Max Duplicate

You already have a fine solution, you've just analyzed it incorrectly.

You do one loop to find the maximum value, and then you do another loop to find duplicates of the maximum value. That isn't a loop inside another loop, that's a loop after another loop -- and that's not O(n) * O(n), it's O(n) + O(n), which is just O(n) again.

No hash tables. No fancy data structures. You already have a good solution.

Well, if you can't use hash table (which makes the problem o(n) time complexity and o(n) space complexity), i would use just sorting the array, and check with 2 pointers if any of the values is repeated. So after sorting the algorithm (O(nlogn)) i would make 2 variable i,j starting i at 0 and j at 1 and while j<arr.lenght, and check if there is duplicate between positions array[i] and arr[j]. Here is how i solved it in js:

function areThereDuplicates(...args){//Don't worry about this, imagine like it is an array
    args= typeof(args[0])=='string'?args.sort():args.sort((a,b)=>a-b)//I sorted the array
    let i=0
    let j=1
    while(j<args.length){
        let el= args[i]
        let secEl=args[j]
        if(el===secEl){
            return true
        }
        i++
        j++
    }
    return false
}