How to add the current project path in Print Composer?

As I investigated in this answer it seems anywhere you can use the Expression String Builder you can use the @project_folder variable in at least QGIS 2.16.

Other potentially useful variables can be found in the annotated screenshot below.

I have not tried this, however it looks like you can create your own custom expression functions (python). This custom expression could be displayed within a composer label. A good how to for creating this would be to go through this blog:

Nathan's QGIS Blog - User defined expression functions for QGIS

Within your function you could use this statement to get your current project document path:

myProjectPath = QgsProject.instance().readPath("./")

A basic implementation that you can add to startup.py:

from qgis.core import QgsProject

@qgsfunction(0, "Python")
def projectpath(values, feature, parent):
    return QgsProject.instance.fileName()

Typing...

[% @project_path %]

...in a QGIS print composer text label will print the file path in the same way as the dynamic text expression does in ArcMap (using QGIS 2.14).