Where is The End
First of all: Yes! There are more "new elements" in reach.
Evidence for elements up to 118 has been reported by Yu.Ts. Oganessian and his collaborations from experiments conducted at Dubna. 48-Calcium (20 protons, 28 neutrons) was and is being used as incoming particle beam, impinging on target foils comprising various actinides ranging from uranium (proton number 92) up to californium (proton number 98). 20+98=118.
Beyond that, it is merely impossible to collect and handle target material with higher proton numbers. Therefore, one is bound to leave the doubly-magic 48-calcium as beam and use, most preferably, something close. That could be 50-titanium with two more protons, i.e. 22; and there we are: 98+22=120.
There are two major problems though:
First of all, the production probability of the fused systems (here with Z=120) decreases rapidly. So instead of observing up to one or two of these superheavy elements per day, one may expect one or two per month, or a handful per year. Such experiments are in principle feasible; for example, a search experiment for element 119 was conducted at GSI last fall, preceeding our 115 experiment. However, such very long around-the-clock experimentsare very demanding on both material and personnel!
Secondly, the lifetimes become shorter and shorter of the superheavy nuclei which can be produced with a given selection of stable ion beams (calcium, titanium, chromium, iron etc.) and actinide target materials. At some point, they do not survive the flight path through the separators, i.e. never make it to the detection set-up themselves! Maybe their decay daughters can be observed, but that poses severe contraints on the discovery.This problem is expected to become a severe problem already for Z=122!
Two solutions: Use more neutron-rich hence radioactive ion beams (future, or in the future, maybe in the future ...), or be smart: Develop new techniques and technologies. Or both. Your pick.
Or give us the money so that we can make a new element and name it after you - or your beloved!
This race is to be continued ...
Moreover, there are theoretical predictions on super- or hyperheavy nuclei with "bubbles" inside or of toroidal shape! Check it out.
Ultimately, there are the neutron stars - ultradense nuclear matter. Look at the sky or check with astrophysics departments.