How many solar systems are there in the universe? How many in the known universe?
According to Wikipedia , "The Milky Way contains between 200 and 400 billion stars and at least 100 billion planets." Then they talk about some vague objects that may or may not count as stars. Every star has something orbiting it, mostly rocks and dust, so let's say about a quarter of them are in the "sweet spot" of being able to hold some of the current definition of "planet."
That's just the Milky Way. There are billions, possibly trillions of galaxies. The Hubble space telescope likes to scope out previously uncharted empty areas of the sky and zero in, and it inevitably sees new galaxies every time it does this. The Milky Way is apparently average-sized, so each of those tiny dots we can only see with the Hubble also has something on the order of half a trillion stars, a quarter of which almost certainly have some planets.
A low guess for the extrapolated universe is 10 to the 24th power solar systems with planets, or 1,000,000,000,000,000,000,000,000, or a trillion trillion. Assuming our solar system is typical, multiply this by eight or nine for the total number of planets. And as we inevitably discover the universe is bigger than we previously imagined, expect that number to go up.