The reason inventory is not increased to its current value is the cost principle, the cost flow assumption, consistency, and other accounting concepts and principles. When a company elects the LIFO cost flow assumption, it chooses to put its most recent costs in the cost of goods sold, and to leave its earlier costs in inventory. The company cannot violate the cost principle by later increasing the inventory to an amount that is greater than those earlier actual costs.
One place that you might find part of the difference between the LIFO cost reported on the balance sheet and its current market value is the "Inventories" footnote to the financial statements. For example, in General Electric's 2011 Annual Report to the SEC (Form 10-K), it indicates that if General Electric had not been using LIFO (for many years), its balance sheet inventory amount would have been greater by $450 million. That does not mean that the difference between its inventory cost and its market value is $450 million, but I suspect that it is part of the difference.