Polar expeditions are the most costly adventures in the world, exceeded only by space travel and deep sea submerging. The main reason is cost of transportation; namely air flights. In addition, the ruggedness of the Arctic Ocean requires expensive customized heavy duty materials and high performance gear. The sled alone is hand made of Kevlar and charges at least 4500 USD (2200 USD for the lighter South Pole version).
The logistic cost of Antarctica is around 140 000 USD per 1-2 person expedition and 250 000 USD for 3-6 (the cost is per payload). The charge includes roundtrip flight to the continent in a large heavy duty plane (Iljusin, Hercules, DC3), a few days stay at Patriot Hills Base Camp, and flight out from the SP in a Twin Otter or Cessna. The 15 minute flight out to the coast starting point (Hercules Inlet) comes extra with 7000 USD for Twin Otter (max 6 pers). Add cost for food, gear, insurance and travel to PA.
Interestingly enough, it comes cheaper to join an ANI guided South Pole expedition. The charge is around 45 000 USD per person; all air transportation, food, gas, gear, 1-3 air food drops and guide included. The reason for this prize discrepancy is another typical ANI mystery.
The North Pole comes cheaper, perhaps due to competition. Unfortunately the skilled veteran pilots of First Air are gone with the airlines recent suspension of services in the area.
Borek Air is now single handedly taking care of business, and prizes have increased although not as high as Antarctica. Only a few years ago the cost was 65000 USD for 5 people with a NP pick up and 2 resupplies. Only 15 grand per person!
The only competition today is from the Russian Borneo Base and their prices too have risen lately. The price example given below is concerning Canada. Count on the equivalent or somewhat less from Russia.
Flight out to the ice (Ward Hunt Island) is 16 000 USD for 820 kg payload. Out from the pole is around 47 000 USD per 725 kg payload, including a stay at Eureka. The prize range differs with regards to your needs. Food drops are extra (below 86.00 degrees 20.000 USD above 86.00 degrees 40.000 USD).
To complicate matters even more, the prices change at both poles with local competition and global fuel costs.
Counting on a budget comes down to sharing flights with other expeditions or travel in a large group to fill up the payloads. But a general budget including all gear and logistics for a solo unsupported expedition is around 100 000 for the North Pole and 150 000 for the South Pole. A two member expedition will come up to around 60 000 USD per person for the North Pole and 100 000 USD per person for the South Pole.