Estimates suggest there are around 2 trillion galaxies in the observable universe.
This number comes from deep-field observations by the Hubble Space Telescope, especially the Hubble Ultra Deep Field (2004) and later refinements using advanced modeling and data from newer telescopes. These studies showed that the universe contains far more faint, small galaxies than previously thought.
However, this is just the observable universe, which spans about 93 billion light-years across. Beyond that, the cosmos may continue—perhaps infinitely—so the total number of galaxies could be vastly greater, or even uncountable, depending on the actual shape and boundaries of spacetime.
Would you like a visual illustration of this cosmic scale?