The standard Pocket Wheel arrives with a durable and maintenance-free sprayed acrylic finish.  However, you may wish to personalize your wheel by ordering it unfinished applying your own wood finish.  While not a trivial task completed in a single afternoon, finishing your own wheel is not difficult.  Here are my suggestions to get the best results.

You will want a well-ventilated and dust-free area, where the wheel parts can dry undisturbed over several days.  Note that oil-based and water-based wood finishes both contain volatile organic solvents, which will evaporate as the finish dries.  Some of these solvents have an objectionable odor or are flammable, so consider your space before starting the project.  While the finish will appear dry after a few hours, best durability is achieved by waiting several days for the final coat to completely harden or "cure" -- the minimum time for this will be listed on the packaging, but is commonly 75-100 hours or 3-4 days.

There are generally two kinds of wood finishes, penetrating and surface-building.  A third kind of finish is paste wax, which is usually applied over another finish to improve the appearance or feel.

Penetrating finishes (such as linseed oil, and "true" tung oil) are excellent for highlighting the beauty of wood grain and can be wiped on with a rag, perhaps four or five coats would do the job. These finishes don’t provide much protection from wear or water, so I’d suggest not using them on a wheel that will see heavy use or be bumped around often in transport. If you go this route, be cautious with the used rags, as the finishes give off heat as they cure -- this heat is enough to start a fire in used finishing rags wadded up in a garbage can.  Note also that "tung oil" has become a generic term for an oil-varnish blend, often not even containing oil from the tung nut.  "True" tung oil is a fairly expensive finish, usually available only at woodworking hobby stores, and will have very prominent labeling of the tung oil percentage.  "Boiled" linseed oil used to be produced by boiling for a few hours to polymerize (make longer molecule chains) the oils and decrease the drying time -- the modern way is to add trace metal driers which heat up the applied finish to speed the curing (this is one reason the rags can self-combust).

Surface-building finishes are typically used on modern furniture, and provide much better protection against wear and water. Shellac and lacquer are two classics that slowly build layers, and although they dry quickly, a dozen or more coats can be needed. Shellac has fallen out of favor with most modern furniture makers, largely because of the time and labor required for a dozen coats on large pieces, but it would make an excellent finish for your wheel. Lacquer uses some pretty volatile solvents, so I can’t recommend using it. Because these finishes dry through evaporation and don’t chemically cure, both shellac and lacquer can be easily repaired or refinished.

A modern surface-building finish is varnish, which is a blend of oils and resins which chemically react with air to make a very tough finish. Of varnishes, polyurethane is the most commonly used for indoor furniture, because it has superior protection qualities and is simple to apply (although it can look plastic-like when heavily applied).  These are usually applied with a brush, but can be thinned for wiping with a rag (thin oil-based by using mineral spirits or naptha VM&P -- initials for "Varnish-Maker's and Painter's").  Commercial "Wiping Polyurethane Finish" is one-half regular oil-based polyurethane varnish, and one-half mineral spirits.  Water-based urethane and acrylic surface-building finishes are also available -- these are usually more clear, but will require more coats than the similar oil-based finishes to obtain the same thickness.

The majority of the finishes found in the hardware stores will be a blend of the above two types -- for example, "Danish Oil" is one-third linseed oil (with metal drying agents added), one-third urethane varnish, and one-third mineral spirits thinner.  It will highlight the wood grain while adding some surface protection, but several coats may cause the finish to look "muddy", as the coats of oil after the first will not be able to penetrate the previous layer of varnish into the wood.

An older finish is oil and wax -- this is a penetrating oil finish which has been polished with one or more waxes (such as beeswax, carnauba palm, lanolin, or paraffin).  In the 1800's, raw nut oil would be wiped onto the wood, then "rubbed" or buffed by hand to better work the oil into the wood's pores and remove any excess.  This was repeated several times a day for up to a month, until the surface was acceptably smooth, then wax was applied and buffed to an appropriate sheen.  A modern application may blend the waxes and oil together to simplify commercial packaging, and use very fine sandpaper to speed the smoothing process.  As noted above, the penetrating oils in an oil and wax finish won't provide as much protection as varnishes, but are simpler to repair and maintain.

Whichever finish you choose to apply, please practice on some scrap wood of the same type.  If you can't obtain any from a local woodworker or cabinet shop, let me know and I'll get you some scraps to "swatch".