how to add new <li> to <ul> onclick with javascript

You were almost there:

You just need to append the li to ul and voila!

So just add

ul.appendChild(li);

to the end of your function so the end function will be like this:

function function1() {
  var ul = document.getElementById("list");
  var li = document.createElement("li");
  li.appendChild(document.createTextNode("Element 4"));
  ul.appendChild(li);
}

First you have to create a li(with id and value as you required) then add it to your ul.

Javascript ::

addAnother = function() {
    var ul = document.getElementById("list");
    var li = document.createElement("li");
    var children = ul.children.length + 1
    li.setAttribute("id", "element"+children)
    li.appendChild(document.createTextNode("Element "+children));
    ul.appendChild(li)
}

Check this example that add li element to ul.

Just use innerHTML:

function function1() {
  ul.innerHTML += `<li> four </li>`;
}

Object.assign can be used to set the textContent (and the id) of the <li> element directly, and the return value can be passed to Element#append to add the list item to the unordered list.

ul.append(Object.assign(document.createElement('li'), {textContent: 'Element 4'}));

Full example:

const ul = document.getElementById('list');
document.querySelector('button').addEventListener('click', e => {
  ul.append(Object.assign(document.createElement('li'), 
              {textContent: 'Four', id: 'element4'}));
}, {once: true});

<ul id="list">
  <li id="element1">One</li>
  <li id="element2">Two</li>
  <li id="element3">Three</li>
</ul>
<button>Add item</button>

This is more easy and straightforward. We don't need to use document.createTextNode unless if we need more flexibility.
Simpler and more practical way is using textContent, which is also a recommended way in JavaScript.

document.getElementById('addItemBtn').addEventListener('click', function() {
    // Get the input field and the list
    const inputField = document.getElementById('itemInput');
    const itemList = document.getElementById('itemList');

    // Get the value from the input field
    const itemValue = inputField.value.trim();

    // Check if the input is not empty
    if (itemValue !== '') {
        // Create a new list item element
        const newItem = document.createElement('li');
        newItem.textContent = itemValue;

        // Add the new item to the list
        itemList.appendChild(newItem);

        // Clear the input field for new input
        inputField.value = '';
    }
});