It also appeared to me that the higher you go, the harder it is to
make an element.
This is pretty much true. I have a tendency to be wordy and long in my posts, but I'll try to cover a few points as concise as possible.
Ununoctium was created by
"bombarding atoms of californium-249 with ions of calcium-48. This
produced ununoctium-294, an isotope with a half-life of about 0.89
milliseconds (0.00089 seconds), and three free neutrons. The
californium target was irradiated with a total of 1.6*10^19 calcium
ions over the course of 1080 hours, resulting in the production of
three atoms of ununoctium."
so, as Curiousone points out, it's very hard, requiring very sensitive equipment and a good deal of patience. Bombarding Californium with a neutron rich Calcium isotope for 45 days, to make 3 atoms and those atoms have half lives of less than 1/11,000th of a second. It's really a matter of having top notch detection devices and knowing exactly what you're doing. If the atoms are bombarded too slow, they won't merge and too fast, there's too much energy so they tend to break apart. The velocity of impact has to be extremely precise.
It's also not a simple progression of difficulty. Specific isotopes require the right building blocks, so some are easier to achieve than others.
Here's a kind of fun article on the creation of element 117. http://mashable.com/2014/05/02/super-heavy-element-117/
Another problem is that the Island of stability (Island of slightly longer half lifes might be more accurate). . . but I digress. As atomic nuclei add protons, the neutron to proton ratio for optimal stability tends to goes up. This throws a monkey wrench into the search for new elements because it's hard to get the right number of neutrons for theoretical optimal stability. See pretty picture.
The Island of stability might exist around 115 protons and 180-182 neutrons. Ununpentium 295, 296 or 297 but Uup is currently created by bombarding Americium 243 with Calcium 48. That's 4 neutrons short of the Island, but there's no good combination of elements that has enough neutrons to reach the 115-180 ratio. Calcium 48 is a good choice because it has 28 Neutrons and 20 protons, a very high neutron ratio, but still less than optimal.
That's why super-novas and possibly neutron star collisions are probably much better at this than we are, cause those processes happen in effectively a dense crowd of molecules and an abundance of available neutrons.
Anyways, What's the use of creating new elements? Has there been any
research on the practical applications of these new elements?
There's nothing wrong with research for the sake of research. The site I linked above, and here is a pretty handy site for quick and easy descriptions of elements and their uses. The most massive element with any practical use is Californium.
Californium-252, an isotope with a half-life of about 2.6 years, is a
very strong neutron source. One microgram (0.000001 grams) of
californium-252 produces 170,000,000 neutrons per minute. It is being
used as a neutron source to identify gold and silver ores through a
technique known as neutron activation. It is also being used in
devices known as neutron moisture gauges that are used to find water
and oil bearing layers in oil wells.
So, there are already some practical uses for man made elements. Americium for example is used in smoke detectors.
There could also be uses in the future for Island of Stability elements if they're ever created. Polonium also has uses.
Is there a limit on how big massive an atom can get?
More protons tends to grow unstable, so there very likely is a practical limit outside of Neutron star pressure where the game changes a bit and the limit could grow into the 200s. The Island of Stability is usually placed in the 110-120 proton range, so we may have already reached kinda close to the maximum number of protons but not the number of Neutrons. If I was to say anything definite here, I'd be guessing, but I wouldn't be surprised if we're not far away from the practical limit.