Kurdistan in Northern Iraq is a challenging but rewarding destination full of history and welcoming Kurdish locals. For a historical and cultural tour check out Karwan Wahed Kurdistan Tours, www.kurdistantours.net.
Or for something more intrepid Secret Compass can take you on a mountaineering expedition to Mount Halgurd, the highest peak wholly in Iraq, near the Iranian Border. Secret Compass is an exceptional leader in taking you on adventures off the beaten path. See www.secretcompass.com.