Can I present a toast when alert is also presented?

Do you know if it's possible to present a toast message while alert is presented?

More specific, the alert contains a textfield and I want to present a toast with error message when textfield's value is over Integer's max value. So, when user gives me an incorrect value, I want to inform the user before he/she presses the alert button.

Can you help me?

There is no such a thing as toast in native UIKit components!

So you can either use another Alert on top of the previous one:

class MainViewController: UIViewController {
    @IBOutlet weak var button: UIButton!
    var alert: UIAlertController!

    @IBAction func buttonDidTouch() {
        alert = UIAlertController(title: "Authentication", message: "Choose a username and password", preferredStyle: .alert)

        alert.addTextField { textfield in
            textfield.placeholder = "Username"
            textfield.addTarget(self, action: #selector(self.changeUsername), for: .editingChanged)
        }
        alert.addTextField() { $0.placeholder = "Password" }
        alert.addAction(.init(title: "Register", style: .default))
        present(alert, animated: true)
    }

    @objc func changeUsername(sender: UITextField) {
        let text = sender.text!
        guard text.count > 5 else { return }
        sender.text = String(text.prefix(5))

        let toast = UIAlertController(title: "Invalid username", message: "The username should not be longer than 5 letters", preferredStyle: .alert)
        toast.addAction(.init(title: "Ok", style: .cancel))

        alert.present(toast, animated: true)
    }
}

_{Note: You may want to get rid of all !s and make them safe as you wish}

Or create a new window with a level more than the alert like the statusBar (or using libraries like this) and present it on top of the current window to achieve something like this:

Note: window management is a large topic and beyond the scope of this question. I suggest you to start with simple approaches and then go for tough ones. You can start with this:

class MainViewController: UIViewController {
    @IBOutlet weak var button: UIButton!

    lazy var window: UIWindow = {
        let w = UIWindow(windowScene: view.window!.windowScene!)
        w.windowLevel = .statusBar
        w.backgroundColor = .green.withAlphaComponent(0.7)
        return w
    }()

    @IBAction func buttonDidTouch() {
        window.makeKeyAndVisible()
    }
}