I've assumed that ids are meant to be unique here. some is a great function for checking the existence of things in arrays:

const arr = [{ id: 1, username: 'fred' }, { id: 2, username: 'bill' }, { id: 3, username: 'ted' }];

function add(arr, name) {
  const { length } = arr;
  const id = length + 1;
  const found = arr.some(el => el.username === name);
  if (!found) arr.push({ id, username: name });
  return arr;
}

console.log(add(arr, 'ted'));

jQuery has a utility function for this:

$.inArray(value, array)

Returns index of value in array. Returns -1 if array does not contain value.

See also How do I check if an array includes an object in JavaScript?

    function countArray(originalArray) {
     
    	var compressed = [];
    	// make a copy of the input array
    	var copyArray = originalArray.slice(0);
     
    	// first loop goes over every element
    	for (var i = 0; i < originalArray.length; i++) {
     
    		var count = 0;	
    		// loop over every element in the copy and see if it's the same
    		for (var w = 0; w < copyArray.length; w++) {
    			if (originalArray[i] == copyArray[w]) {
    				// increase amount of times duplicate is found
    				count++;
    				// sets item to undefined
    				delete copyArray[w];
    			}
    		}
     
    		if (count > 0) {
    			var a = new Object();
    			a.value = originalArray[i];
    			a.count = count;
    			compressed.push(a);
    		}
    	}
     
    	return compressed;
    };
    
    // It should go something like this:
    
    var testArray = new Array("dog", "dog", "cat", "buffalo", "wolf", "cat", "tiger", "cat");
    var newArray = countArray(testArray);
    console.log(newArray);