drawing lines in Illustrator

If you click on the other paths in the problem document, can you see the anchor points? That is, is the problem document wide?

Try using the Pen tool and hold down the Cmd/Cntl key to access the Direct Selection tool and the Opt/Alt key to access the Anchor Point too. 

Also check your preferences to change the size of the anchor points. 

You cannot draw with the direct selection tool. Press P for pen and click the end point.

Hi ​@L0KiTo,

If you copy and paste that line into a new document, can you continue adding onto the line there? Can you add a new anchor point with the pen tool elsewhere on the line, and remove the end node?

If the line works properly as expected in a new document, try saving your working file, closing it and restarting the document (or even Illustrator). If that doesn’t help, and it’s not too difficult, try copying everything in your file into a new document (could be a glitch or a setting in that document), and this will at least help you move forward.