The answer will vary a bit depending on whether you are simply interested in going from port to port, or if you have specific destinations in mind.
RTW by Ocean Liner
Many passenger lines have vessels that operate around the world or close to it, on a set itinerary, and a web search for "around the world cruise" will turn up options from Cunard, P&O, Princess, Silver Sea, Regent, and so on. I didn't see any that covered both poles, but five oceans and the Amazon and the Nile could be possible. Holland America operates a 113-day "Grand World Voyage" stopping at around 40 ports, on which a "Deluxe Verandah Suite" starts at US$155,200 for two. This seems to be the sort of trip Richie Tanenbaum was on.
RTW by Freighter
As an alternative, however, one can arrange for passage on cargo ship. Someone looking for peace and quiet on the ocean may not find it on an ocean liner with a thousand other passengers, or on a cruise ship with two thousand other passengers, but passage on a freighter means there will probably be no more than a dozen passengers total, plus the crew.
I would not be deterred by the dismal depiction of onboard life in The Life of Pi; the quality of accommodations and amenities will vary by ship. No freighter will be mistaken for the Queen Mary II, but then, you can find round-the-world freighter voyages for under $15,000. SeaPlus.com is a guide for freighter travel, and Flightless Travel's Cargo Ship page has some useful links.
RTW by Sailboat
Since we're on the topic of water travel around the globe, I'll add a note on sailing trips. A sailboat trip around the world can take years, and unlike the other two options requires your active participation and physical hardship, so you can't just walk up and buy passage with just anyone. There are several outfits like Clipper Round the World, however, that will take people with no experience and train them. If you're an experienced mariner and have your own boat, you could also look into "cruising rallies" organized by groups like the World Cruising Club.