It will climb...and climb, and climb,
"The GA pitch mode initially commands a go-around attitude and then transitions to speed as the rate of climb increases. This speed is normally between command speed and command speed + 25 knots."
My take on it is that you will never capture the altitude in the MCP window because it is below you.
When you select GA, as stated in the quote above (from the FCTM page 4.51), the pitch mode is GA. With AT armed or engaged, the AT mode is GA. The jet will pitch toward GA attitude, and operate with the AT toward achieving a 2,000 fpm climb. If you do nothing other than select GA in these conditions, you will continue climbing until you decide to select something else on the MCP.