One thing which can be said with absolute certainty is that no species in the Universe (or even all their combined efforts) will even come close to exhausting the daunting enormity of chemical space. When you drag combinatorics into a problem, you can easily stumble on massive numbers, and this is the case here; it has been roughly calculated that there are on the order of $10^{60}$ "chemically reasonable" distinct molecules below a molar mass of $\mathrm{500\ g\ mol^{-1}}$! This means that almost all chemical substances which can exist, never will exist, be it from natural or artificial production.
No matter how far chemistry develops, we will always be interested in investigating the chemical composition of new samples, and will always find many new compounds doing so. Most of the time we'll find the same of what we do on Earth; there's no escaping the fact that $\ce{N_2}$ and $\ce{Al2O3}$ are very stable substances, for example, and can exist in a wide range of conditions. But there will always be something new for us waiting out there. Organic substances are tremendously varied thanks to the concatenability of carbon atoms. One might think inorganic substances are far less variable, but just the class of silicate minerals is enormous, and many polymorphs of the same substances can exist.
Having said that, at least one area of chemistry seems far closer to being exhausted; the elements themselves. There is no possibility of us having "missed" any element among the first 112, and we're still filling in the gaps at higher atomic numbers, but each heavier element discovered involves ever more complicated procedures and lower yields. Much of the experimentally determined properties of elements above $\mathrm{Z=104}$ come from the analysis of a tiny amount of atoms, as little as tens of them. Each new element is significantly more difficult to synthesize, and is less stable, which is hampering progress. We don't yet know very well how to get past $\mathrm{Z=122}$ or so even in principle, and are unsure how accessible the ultraheavy elements will prove to be. There is still great academic interest in several aspects of the elements and their isotopes, but current research deals with the production of very shortlived species, which have limited applicability.
In all, we will always find something new to explore, but whether we will always find something strange depends on what you mean by the word. There are still many, many substances to find which will display impressive biochemical properties, for example, and many lethal or crippling diseases today might still become curable in the future. But there is zero chance of finding a substance which makes magic real, which unlocks faster-than-light travel, which has endochronic properties or which can be used to make overunity engines and perpetual motion devices. Ultimately, Chemistry is a form of applied quantum electrodynamics, a subsection of Physics, and quantum electrodynamics is a very developed theory with incredible precision. Thus, there is little room for surprises, and they likely will not come from a chemical system unless it is very specially tailored, rather than randomly found in nature.
