We have stacked the X-ray emission from X-ray-detected sources in ∼ 3.8 Ms of Chandra Deep Field and XMM-Newton Lockman Hole observations. The fraction of the X-ray background which can be resolved into point sources is ∼ 70-90% up to 4 keV. In the 6-8 keV band this fraction decreases to ∼ 50-70% and XMM-Newton indicates only ∼ 40-60% in the 8-12 keV range. The unresolved fraction is consistent with an undiscovered population of highly obscured AGN. To attempt to uncover this population we proceeded to stack the X-ray emission from the X-ray-undetected optical sources in the Chandra Deep Fields, using Hubble source positions from the Great Observatories Origins Deep Survey. In the 0.5-6 keV range the stacked-source emission corresponds to the remaining 10-20% of the X-ray background. In the 6-8 keV band we find upper limits to the stacked X-ray flux consistent with the remaining ∼ 40% of the background, but the errors are too large to claim a detection of the obscured AGN population.