Greeks were one of the earliest civilizations who had focused on every aspect of life systematically. They always gave importance to nature and it's interaction with people. They discovered or invented many things in BC period and many of them are used even today. Since you asked 10 of such inventions, I will write in a pointwise manner:
The Hippocratic oath we take today was the earliest text of ethics for the modern world. Born in 460 BC in Cos, Hippocrates was the first to study, collect data and experiment to understand the diseases. Although there were superstitions about diseases as a God's punishment, Hippocrates was the first one to oppose and start a scientific approach. This marked a new era and laid foundations for modern medicine. He is referred as the Father of Medicine along with Charaka (Indian personality who developed ayurvedic medicine).
Without the anchor, we cannot imagine a shipping industry. This is one of the historic inventions by Greeks which is used until today. The first anchor was invented by Eupalamas and it was a wooden anchor. The metallic anchor was invented by Anacharsis (592 B.C) with the central idea drawn from the design of Eupalamas.
Perhaps the most widely practiced system throughout the world. The idea of 'every citizen has equal rights' has its roots dated back to 508 B.C is Athens, Greece. Greeks used to conduct meetings which consisted of kings, ministers, council members etc who used to take important decisions regarding the methodology of the rule, implementation of new systems, taxation, wars etc. Everyone's opinion was considered in these meetings and this marked the transition of a political system beginning the first systematic democracy.
4. Alarm Clock
One of the notable inventions from ancient Greeks which we find it everywhere now. It was invented by Ctesibius around 200 B.C. This has undergone several modifications with the modernizing world which came to the existence of what it is now. Greek mathematician Plato modified this alarm and used it wake up in the mornings.
Greeks were the first to mark the navigation in the form of pictures (Cartography) which can be used as a reference. They used it to use as a navigation and travel guide. This methodology is widely practiced in the modern world. Anaximander was the first to publish a world map.
One of the most popular games which have a huge craze in the present modern world was first conducted in 776 B.C in Olympia, Greece. These sports were conducted to impress or please the ancient Greek Gods, Zeus specifically and were conducted once in 4 years.
It is a device used to measure the distance traveled by a vehicle. Though there is an ambiguity about who invented this by the modern historians, everyone points its origin to ancient Greece. They used to calculate the distance they traveled in ancient times so as to have exact calculations with time.
8. The concept of Philosophy
Greeks were the first to question the facts of life swaying away from superstitions, myths, and magic. Most of the Greek philosophical theories are widely in use today. Notable philosophers include Socrates (Father of philosophy), Aristotle etc.
Though the concept of geometry was practiced widely around the world other than Greece, most of them were baseless and just mere assumptions. Greeks were the first to question the assumptions of all the geometrical principles that was practiced in other countries. They were the first to introduce the theoretical explaination for existing geometrical principles which is widely followed around the world. Most notable mathematicians who introduced geometrical principles are Thales (father of geometry), Pythagoras, Euclid, Archimedes etc
The art of theatre or stage performances was introduced to this world by Greeks. They were used by them to show how Gods created humans, how they aided humans in great wars and also used them as a means of education for not only to the citizens of Greece but also to people of other nations who visited them. This art diversely transformed into modern day cinemas and other forms of arts.
Dr. V. Shashikanth