There is not an easy answer to this question. First what do you mean by a word? When you have a 300 word writing assignment to hand in as homework you count anything separated by spaces. However let me give you a few more things to think about.
Is the really a word or should we only count things (nouns), actions (verbs) and descriptions (adjectives)?
If tall is a word, are taller and tallest other words or different forms of the same word?
Is Hong Kong one word or two? Is the short form HK another word?
Is a phrasal verb like throw up (to vomit, not to throw vertically) a different word from throw and up?
Is newspaper a different word from news and paper?
Is the flash in flash drive, the same word as in flash photography?
Is bow (with two pronunciations) one word or several? As a verb it can mean to bend the middle of your body to show respect, to kneel on the ground, or just to lower your head. As a noun bow is the act of bending the middle of your body, the front of a ship, a weapon made from a long piece of wood under tension which is used for shooting arrows, a decorative knot tied with a ribbon, a long thin tool used for playing stringed instruments such as the violin.
Do we count the many different words for one thing? Milk sugar is the type of sugar that naturally occurs in milk. Milk sugar is a combination of two other sugars galactose and glucose and so is called a disaccharide. Milk sugar is often called lactose by biologists and doctors. Physical chemists call it C12H22O11. Organic chemists call it β-D-galactopyranosyl-(1→4)-D-glucose. How many words should we count for this one thing?
Hamburger is a German word for people and things from the German city of Hamburg. It is also an English word for a food that came from that city. Originally hamburgers, the food, were called "Hamburg steak". Dim sum is an English word of Cantonese origin (See the Dictionary of Hong Kong English), but it is only understood by a minority of English speakers. Should both hamburger and dim sum be counted as English words? Is dim sum one word or two?
Is senight a word? It is an older word for a week or half a fortnight (2 weeks). Only a minority of English speakers understand the word fortnight, even fewer understand the word senight. Should we count older words that are rarely used?
If we count every entry in a dictionary, which are called headwords, we only get part of the answer. There are probably about a million words in English, but it depends on what you count.