Per the docs, a Liveagent Chat button/link can be configured to route chat requests to agents using one of three methods: Choice, Least Active, and Most Available. Least Active and Most Available are push methods, pushing the chat to an agent. When using these methods, you can set a push timeout value; if the agent doesn't response within the timeout value, the chat is pushed to another agent.
Per this Knowledge article, when using a push routing option, if a chat request is routed to each available agent in turn and no one takes the chat, the requesting user will be informed that no one is available to chat.
Question: When using Choice routing, chat requests are not pushed; agents see the chat request appear on screen and must click accept. If no agent accepts the chat, will the request ever time out? Will the requesting user ever receive notification that the chat request cannot be completed? The system will not allow you to set the push timeout value when using Choice, and initial tests suggest that the chat window will wait indefinitely, as long as there is at least one Liveagent user with the required skills in the Online state.