Various Sleeping Strategies were discussed on the page in the Mental Approach & Strategy section of this website. This pages focuses on what equipment you might choose to bring.
The amount of sleeping equipment carried by bikepackers varies as much as their sleeping strategies. Even for people who plan to sleep in a hotel every night, I strongly recommend having a backup plan in case you arrive in a town and all the hotels are full or if you are too tired to go further and there are no hotels for a long distance. The minimum would be to bring an emergency blanket, which weighs less than 100 grams and costs almost nothing.
An emergency blanket should ensure that you can survive a night outdoors during the summer if necessary without any serious problems, but if sleeping outside is part of your plan then I recommend bringing a small amount of sleeping gear to make the experience more comfortable. At the other end of the extreme from a simple emergency blanket is bringing a tent, sleeping bag, sleeping mat, and pillow. Many people delete one, two, or three of these items depending on the degree of comfort that they desire and can tolerate. Most setups weigh between 0.5 and 1.5 kg, so will cost 20-60 minutes in riding time over the complete route (see the Weight page), which could easily be gained back by being better rested.
I used a very small and lightweight tent for my first race in 2014, but was frustrated by the amount of time that it took to assemble and repack, and so sometimes only used it as a groundsheet to lay on top of. In later races, I used a bivvy bag, a Helium by Outdoor Research (Amazon), that is far quicker to setup and is even easier to find a place to stealthily wild camp with. I like the Helium bivvy because it has an optional pole to hold the material away from my face, gives decent protection from the elements and breathes reasonably well, but there are many other good options, see here.
To ensure a comfortable rest, I use an inflatable mat with an integrated pillow, the Klymit Inertia X-Wave (Amazon), whose best attribute is how compact it is when deflated. The sleeping bag that I use in the summer is very minimal and weighs under 400 grams, an OMM Mountain Raid 1.0 (Amazon). I can always wear more clothes and if I expect it to be particularly cold then I’m likely to search for a hotel instead.
Some bikepackers take a bivvy bag with nothing to go inside or only something tiny like a sleeping bag liner. Other people leave the bivvy bag / tent at home and only bring a sleeping bag, and then search for somewhere covered or don’t sleep outside when it’s wet. Another option is a hammock, which some people find to be very comfortable, but is more limited in where it can be used. There are multiple options of what to use for a pillow, including extra clothes, a bike helmet, etc.
The final page in this Accessories & Equipment section contains estimates of Total Equipment Costs.