I remember having similar confusion learning about analog electronics the first time. The book and my calculator would show some value, and the real-world circuit on a breadboard would be a little different.
It's convenient to think of those black lines on a schematic as a perfect conductor, a perfect wire. But in reality, even the wires connecting components together have some small resistance. It doesn't matter if they're "oxygen-free copper" or solid gold or even a very expensive super-conductor chilled to extremely low temperatures: they will all have some small resistance.
If somehow you could achieve a perfect "zero ohm" resistance, the current would necessarily have to approach infinity, which is at best an abstract concept. But let's say you found the world's most hefty power supply. You would only be finding ways to vaporize your low resistance wires and components...
Current equals voltage divided by resistance. If resistance is zero, then you're trying to divide by zero, which is an error.* 1 volt across 1 Ω is 1 amp. Across 1mΩ would be 1 kA. Across 1µΩ would be 1 MA. The current is inversely proportional to resistance. In the real world you encounter limits of equipment and components well before these hypothetical large current values occur.
* Unless you are Chuck Norris. :)