I am going to be trying to add an electrical circuit to a gazebo that
is being placed in a backyard that is already completed/grown out. I
don't think it would be possible to dig a trench without compeltely
destroying portions of our plantings, not to mention the cost of
trenching. Is there a type of conduit that can be placed above ground
(its about a 50' run) and safely house electrical from the elements?

If you're going to have something above ground, might as well stick with
an extension cord. Even if conduit is allowed above ground, it would be
ugly and vulnerable to damage.

50' is not a long distance, and it won't cost anything other than a few
days of digging. I've hand dug hundreds of feet of trenches on our
property for power lines, water pipes, drain lines, etc., and we have
very rocky soil. If you take the sod up before digging, you can easily
rake the dirt back in and put the sod back in place afterwards. Or, just
dig the trench and throw on some grass seed afterwards. A few weeks
later, you won't even know the trench was there.

Your trench doesn't have to be a straight line, you can navigate around
trees or other plantings as needed. You can even tunnel under large
roots, pipes, sidewalks, etc.

If you don't plan on replacing the cable in the future, you can skip the
conduit and use direct burial UF (underground feeder) cable. You'll
probably have to dig the trench a little deeper, but around here bare
cable only needs to be 6" deeper than conduit (24" vs 18").

If your health or physical condition does not allow you to hand dig a
trench, you could probably hire a neighborhood kid to do the digging for
you.

If you're really concerned about the impact on your yard, or the time it
takes to dig the trench, you might want to check around for someone with
a "vibratory plow". They simply hook a cable or pipe to the machine, and
drive it through your yard. The phone and cable companies around here
use them all the time, and the plow line is practically invisible when
they are done. The guy that pulled my cable line (300') told me his walk
behind machine could pull up to 1" pipe up to three feet deep. It's a
great option if you don't have a lot of rocks, roots, or other
obstructions.

Regardless of what option you choose, remember to "call before you dig".
And whenever possible, try to keep a map of all the underground cables
and pipes you install. I always take lots of photos, and have found them
very helpful when I need to do new work at a later date.

Good luck,

Anthony