One thing that annoys the living crap out of me is when live chat is the only way to get support and they have office hours! So now I can't send an email at 1 AM when I have some time to myself, but have to take time out of my day to chat with customer support.
Some live chat systems will create a ticket from your off hours chat, and then followup with an email once their support team comes back online. I've even seen when this email allows to click a link to go into a new chat session (assuming on hours).
I might as well use the bloody phone by then.