In the beginning stars only consisted of the hydrogen element and due to nuclear fusion of those elements in stars and supernova's more heavy elements were created. Because of that, like our Sun, the stars were made not only of hydrogen but also oxygen and carbon etc.
Now is it expected that in future more stars will be made of more heavy elements or are there causes/laws which prohibit stars forming of e.g. stars made of elements without hydrogen.
The initial stars were made of hydrogen and helium.
These enriched the interstellar medium (ISM) with some chemical elements right across the periodic table, when massive primordial stars ended their lives as supernovae.
Subsequent generations of stars continue to enrich the ISM, if their lives are short enough.
So the general gist of what you suggest is true, but the rate of enrichment is very slow. The Sun was born 4.5 billion years ago, but sun-like stars born within the last few million years also have a metallicity consistent with the Sun.
Why is this? Well, star formation is quite inefficient; only about 10 percent of a collapsing gas cloud ends up in stars. Then a star only processes about 10-20 percent of its mass as nuclear fuel; and then only a fraction of this is returned to the ISM in the form of winds and supernova ejecta.
To get a more enriched ISM requires bursts of massive star formation in dense environments. Such conditions occurred early in the life of the Milky Way and perhaps will again when we collide with the gas-rich Andromeda galaxy.
In conclusion, star formation will continue into the future; the stars will very slowly get more metal-rich; there is nothing to prevent more metal-rich stars forming; perhaps the main consequence would be a small reduction in the maximum possible stellar mass (due to the increased gas opacity and consequent radiation pressure).
Just to add, while I think Rob Jeffries answer covers it.
Now is it expected that in future more stars will be made of more heavy elements or are there causes/laws which prohibit stars forming of e.g. stars made of elements without hydrogen.
While this is unlikely to happen because Hydrogen will stick around as the most abundant element for a very very very long time. It will exist in gas clouds, and expelled by stars in coronal mass ejections and stellar nebulas from stars that are too small to go supernova and gas giant planets and very slow burning red dwarf stars, so a star and a galaxy without hydrogen is unlikely.
That said, in theory, without hydrogen stars wouldn't burn neatly at all. Hydrogen is unique in the way it can fuse slowly in the core of a star and at comparatively low temperature. Helium fusion in smaller stars is a associated with a helium flash and carbon fusion in smaller stars is basically a type 1a supernova. Very large stars could fuse helium and carbon for a relatively short stellar life-span but stars of that size are rare. Smaller stars are much more common, and without hydrogen, they wouldn't behave like stars.
The increase in heavy elements, called metallicity of stars, does slowly increase over time. It increases faster in the center of galaxies than the spiral arms. Our sun for example contains about 2% by mass heavy elements. Metallicity in this case means carbon or heavier elements, not metals.
