Most visitors to Askja and Kverkfjöll come on organized tours, which makes sense, given the poor roads and other logistical hassles. However, except for the infrequent hiking trips led by Ferðafélag Akureyrar and Ferðafélag Fljótsdalshéraðs, no tours allow extensive time for exploring either destination. For Askja, Mývatn Tours is usually willing to leave you at the Drekagil hut and pick you up a day or two later. The SBA-Norðurleið tour allows 2 nights and a day at Kverkfjöll, but for more time you'll need your own 4WD vehicle.
Mývatn Tours (tel. 464-1920; www.askjatours.is) leads a popular 11- to 12-hour guided tour of Askja, starting from the Mývatn information center, for 11,000kr ($176/£88) adults and 5,500kr ($88/£44) for children 6 to 12. Departures are daily from July 10 to August 20, and three times weekly in late June, early July, and late August. Bring lunch, warm clothes, strong shoes, and a bathing suit and towel. Hótel Reykjahlíð in Mývatn (tel. 464-4142; www.reykjahlid.is) offers roughly the same tour, but in Super Jeeps instead of a bus. The cost is 14,800kr ($237/£118) for adults and 9,500kr ($152/£76) for children 6 to 12, with a four-person or 59,200kr ($947/£474) minimum. Off-road driving is prohibited, however, so the much higher price is hard to justify. Departures are from June 15 through September by advance request.
From early July to late August, SBA-Norðurleið (tel. 550-0700; www.sba.is) offers a 3-day trip from Akureyri or Mývatn to Askja and Kverkfjöll, with 2 nights at Kverkfjöll's Sigurðarskáli hut. Departures are Mondays only, and the 21,500kr ($344/£172) price (half-price for children 12-15) includes a guide but no food or accommodation. If you ask, they can at least reserve a bed for you at the hut.
In July, Ferðafélag Akureyrar, Akureyri's hiking organization, leads an adventurous 7-day backpacking trek (53,000kr/$848/£424 per person, including accommodation, food and guide) from Herðubreið to Askja and onward to Svartárvatn lake, at the southern terminus of Route 843. The website is in Icelandic only; find the "Ferðaáætlun" link, and scan the tour listings for "Öskjuvegur." A 3-day trip to climb Herðubreið should also be listed. Ferðafélag Fljótsdalshéraðs occasionally leads hiking trips in the Askja-Kverkfjöll region (click on the "ferðir" link to glean tour listings, or just call instead.).
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.