ListBox items return string, when DataTemplate is Button

You need FrameworkTemplate.FindName Method (String, FrameworkElement) for this purpose:

private childItem FindVisualChild<childItem>(DependencyObject obj)
where childItem : DependencyObject
{
        for (int i = 0; i < VisualTreeHelper.GetChildrenCount(obj); i++)
        {
            DependencyObject child = VisualTreeHelper.GetChild(obj, i);
            if (child != null && child is childItem)
                return (childItem)child;
            else
            {
                childItem childOfChild = FindVisualChild<childItem>(child);
                if (childOfChild != null)
                    return childOfChild;
            }
        }
        return null;
}

Then:

for (int i = 0; i < Game_ScrollViewer_online.Items.Count; i++)
{
     ListBoxItem GameListBoxItem = (ListBoxItem)(Game_ScrollViewer_online.ItemContainerGenerator.ContainerFromIndex(i));
     ContentPresenter contentPresenter = FindVisualChild<ContentPresenter>(GameListBoxItem);
     DataTemplate myDataTemplate = contentPresenter.ContentTemplate;
     Button tempBut = (Button) myDataTemplate.FindName("Current_game_button", contentPresenter);
     //Do stuff with button
}

To solve missing FindName use FindDescendant like this:

public T FindDescendant<T>(DependencyObject obj) where T : DependencyObject
{
    if (obj is T)
        return obj as T;

    int childrenCount = VisualTreeHelper.GetChildrenCount(obj);
    if (childrenCount < 1)
        return null;

    for (int i = 0; i < childrenCount; i++)
    {
        DependencyObject child = VisualTreeHelper.GetChild(obj, i);
        if (child is T)
            return child as T;
    }

    for (int i = 0; i < childrenCount; i++)
    {
        DependencyObject child = FindDescendant<T>(VisualTreeHelper.GetChild(obj, i));
        if (child != null && child is T)
            return child as T;
    }

    return null;
}