If you only want to run the function given to useEffect after the initial render, you can give it an empty array as second argument.

function MyComponent() {
  useEffect(() => {
    loadDataOnlyOnce();
  }, []);

  return <div> {/* ... */} </div>;
}

As mentioned in the comment, there is no point in making the useEffect hook dependent on the count, if you are not going to update the title every time the count changes. The code could be:

const UseEffectCounter1 = () => {
    const [count, setCount] = useState(0);
    useEffect(() => {
        document.title = 'You clicked 0 times';
    }, []);
    return (
        <div>
            <button onClick={() => {setCount(count + 1)}}>+1</button>
        </div>
    )
}

Anyway, if you really want to avoid the hard-coded value, you could do something like:

const UseEffectCounter1 = () => {
    const initCount = 0;
    const [count, setCount] = useState(initCount);
    useEffect(() => {
        document.title = `You clicked ${initCount} times`;
    }, [initCount]);
    return (
        <div>
            <button onClick={() => {setCount(count + 1)}}>+1</button>
        </div>
    )
}

As initCount is never going to change, the useEffect hook will be triggered only once.

You can use more than one useEffect().

For example, if my variable is data1, I can use all of this in my component:

useEffect( () => console.log("mount"), [] );
useEffect( () => console.log("data1 update"), [ data1 ] );
useEffect( () => console.log("any update") );
useEffect( () => () => console.log("data1 update or unmount"), [ data1 ] );
useEffect( () => () => console.log("unmount"), [] );

There are a couple of things there. First, to fix the code, you could update your useEffect to this:

useEffect(() => {
    messagesRef.on('child added', snapshot => {
    const message = snapshot.val();
    message.key = snapshot.key;

    setMessages(messages.concat(message)); // See Note 1
}, []); // See Note 2

Note 1

The setMessages line is how you update your state. useState is a little bit different from the "old" setState in a sense that will completely replace the state value. React documentation says:

This is because when we update a state variable, we replace its value. This is different from this.setState in a class, which merges the updated fields into the object.

Note 2

React Hooks changes the way we build apps and it is not a real "translation" from the old lifecycles.

The empty brackets ([]) in the last line, will make your code "similar" to componentDidMount, but most importantly, will make your effect run only once.

Dan Abramov said (removed some of the original text):

While you can useEffect(fn, []), it’s not an exact equivalent. Unlike componentDidMount, it will capture props and state. So even inside the callbacks, you’ll see the initial props and state. (...) Keep in mind that the mental model for effects is different from componentDidMount and other lifecycles, and trying to find their exact equivalents may confuse you more than help. To get productive, you need to “think in effects”, and their mental model is closer to implementing synchronization than to responding to lifecycle events.

Full article about useEffect here.