In Lawsuit Against a Sole Member LLC, is the Liability of the LLC Only Equal to the Amount of the Initial Financial Contribution?

No, the Limited Liability Company is liable for all of its own debts, regardless of the amount of capital contribution paid by its Members, even a sole Member.

You may be confusing your thought process with the fact that a Member is not personally liable (in most cases) for the debts of the L.L.C.

Setting aside the ability of a creditor to collect a judgment against an LLC, the LLC itself is liable for any obligation that it contracted for or otherwise owes or incurs, and that judgment could be up to a nearly infinite amount.

The Sole Member who contributed the capital to start the LLC would be at risk of losing that initial contribution (and any loans or additional capital he may have contributed to the LLC) should a judgment be obtained against the LLC and ultimately collected.

Only the LLC is responsible for the debts it creates, not the Members; but the initial investment and additional capital investments could be lost through claims made and won against the LLC.