You can easily adjust the chat slots in the Client.
Under Settings, you can adjust the number of slots under Messages.

You can also adjust the slots during an ongoing chat. If you have reached your maximum number, but feel that you have the situation under control, you can easily adjust the + button that appears at the top left, above the external chats. Similarly, if you have a more demanding chat and can't take any more (despite a higher set maximum number), you can click the - button to lower the maximum number. When you reach the maximum number of slots in the chat, the bar turns red.