Tables are typically no more than a few parts, held together with dowels or tenons. Locate some spare parts from a second-hand table, add some dowels and you've got everything you need to extend height to a table. A portable table might even have removable extensions for convenience. It's basic woodworking that almost anyone can accomplish.
Table height is most often either chair or stool height. Dining table height is standardized at 30 inches to accommodate chair seats that are about 18 inches. Work and craft tables often rise to 36 inches to accommodate stool seats that are 30 inches tall. In some instances, tables even rise to 42 inches to accommodate standing up and working.
Search local flea markets, used furniture stores, or even antique shops to find a table that might suffice for parts. Lots of legs are similar in diameter or design, and might work well for your extensions. Online dealers handle spare table parts, in a multitude of designs and species. It's a safe bet that you can find spare legs that are just what you're looking for.
Cut and Match
Figure out how much length you need and use a miter saw to cut extensions from the second-hand legs, or legs that you've ordered online. Depending on how they match, you might be able to cut all four extensions from a single leg.
Drill and Dowel
Use a drill-press and 3/4-inch sawtooth bit, to drill 3/4-inch, matching holes in the extensions, at least 3 inches deep. Turn your existing table upside down, and remove any hardware from the bottom of the legs. Use a drill/driver equipped with the sawtooth bit to drill matching holes in the bottoms of the legs.
Glue and Pound
Apply wood glue to all of the holes. Insert 3/4-by-6-inch dowels in the four extensions. Insert the extended ends of the dowels into the holes in the legs. Use a hammer to pound them into the legs. Turn the table upright. Allow the glue to dry overnight.
Dowel screws are an option to wooden dowels. Dowel screws are not considered as strong as wooden dowels, but are more convenient and easier to use. Dowel screws look like bolts with pointed threads at both ends.
Install Dowel Screws
Drill matching holes in your extensions, and the bottom of the table legs. Use locking pliers to screw the dowel screw halfway into the bottom of the leg. Screw the extension to the exposed end of the dowel screw on the table leg.
Depending on what your intentions are -- a worktable for example -- you might want a sturdier extension. Four-by-four posts are the solution. It won't be as pretty, but it will support an elephant, and they're removable.
Measure and Cut
Measure and use a miter saw to cut extensions from a 4-by-4 post.
Draw a Footprint
Measure the thickness, or diameter at the bottom of the table legs. Using the measurement, draw a footprint of the leg on the end of one of the extensions. Measure the width of the outside perimeter of the drawing. Use a table saw to rip 1 1/4-inch-thick pine to the measurement.
Make an Enclosure
Measure and cut four pieces of 1 1/4-inch pine, to frame the drawing of the footprint on the end of the extension. The bottom of the leg should fit snug into the footprint.
Screw it On
Drill two holes, evenly spaced, into each of the four pieces of pine with a 3/16-inch bit and drill/driver. Apply glue and screw them to the extensions with 2-inch screws.
Place the Table
Lift up the table. Insert one extension under each leg, allowing the bottoms of the legs to settle into the pine enclosures.
Maybe the easiest solution to gain table height is the addition of a finial. This type of ready-made leg hardware attaches to the bottom of the leg to add a few inches. They come in different shapes and length. Choose one you like, tip the table upside down, drill pilot holes in the ends of the legs and screw them on.