Sure, the most iconic river in all the land is the Colorado that flows through Grand Canyon National Park.

NPS photo.
Indeed, so hallowed is the Colorado that the waiting list for private trips is measured in years, plural.
And why not?
Great white water, that gigantic rift in the Earth that you float through over a period of two or more weeks, the simple thought of retracing Major John Wesley Powell's exploration of the uncharted Western territories in the late 1800s.
Of course, you can avoid that long wait by signing on with one of the permited commercial outfitters that runs the river.
Trips range from as short as one day to as long as 25 days!
Here's the park's list, as of 2024, of permited river outfitters:
Aramark-Wilderness River Adventures
P.O. Box 717, Page, AZ 86040
800-992-8022; 928-645-3296; FAX 928-645-6113
Arizona Raft Adventures, LLC
4050 E. Huntington Drive, Flagstaff, AZ 86004
800-786-RAFT; 928-526-8200; FAX 928-526-8246
Arizona River Runners, Inc.
P.O. Box 47788, Phoenix, AZ 85068-7788
800-477-7238; 602-867-4866; FAX 602-867-2174
Canyon Explorations/Canyon Expeditions
P.O. Box 310, Flagstaff, AZ 86002
800-654-0723; 928-774-4559; FAX 928-774-4655
Canyoneers, Inc.
P.O. Box 2997, Flagstaff, AZ 86003
800-525-0924; 928-526-0924; FAX 928-527-9398
Colorado River & Trail Expeditions, Inc.
P.O. Box 57575, Salt Lake City, UT 84157-0575
800-253-7328; 801-261-1789; FAX 801-268-1193
P.O. Box 0, Kanab, UT 84741
800-544-2691; 435-644-2691
Grand Canyon Whitewater
PO Box 2848, Flagstaff, AZ 86003
800-343-3121; 928-779-2979; FAX 928-779-2973
Hatch River Expeditions
HC 67-35 Marble Canyon, AZ 86036
800-856-8966; 928-355-2241; FAX: 928-355-2266
P.O. Box 67, Angles Camp, CA 95222,
800-346-6277; 209-736-2924; FAX 209-736-2902
Outdoors Unlimited
6900 Townsend Winona Road, Flagstaff, AZ 86004
800-637-7238; 928-526-4546; FAX 928-526-6185
Tour West, Inc.
P.O. Box 333, Orem, UT 84059
800-453-9107; 801-225-0755; FAX 801-225-7979
Western River Expeditions, Inc.
7258 Racquet Club Drive, Salt Lake City, UT 84121
800-453-7450; 801-942-6669; FAX 801-942-8514