I am trying to find the simplest example of a custom widget being written for Gtk-3.
So far the best thing I've found is this (using PyGTK), but it seems to be targeted to Gtk-2.
BTW: I don't care the language it is written in, but if we can avoid C++, much better!
Python3 Gtk3 it is, then:
from gi.repository import Gtk
class SuperSimpleWidget(Gtk.Label):
    __gtype_name__ = 'SuperSimpleWidget'
Here is a non-trivial example that actually does something, namely paints its background and draws a diagonal line through it. I'm inheriting from Gtk.Misc instead of Gtk.Widget to save some boilerplate (see below):
class SimpleWidget(Gtk.Misc):
    __gtype_name__ = 'SimpleWidget'
    def __init__(self, *args, **kwds):
        super().__init__(*args, **kwds)
        self.set_size_request(40, 40)
    def do_draw(self, cr):
        # paint background
        bg_color = self.get_style_context().get_background_color(Gtk.StateFlags.NORMAL)
        cr.set_source_rgba(*list(bg_color))
        cr.paint()
        # draw a diagonal line
        allocation = self.get_allocation()
        fg_color = self.get_style_context().get_color(Gtk.StateFlags.NORMAL)
        cr.set_source_rgba(*list(fg_color));
        cr.set_line_width(2)
        cr.move_to(0, 0)   # top left of the widget
        cr.line_to(allocation.width, allocation.height)
        cr.stroke()
Finally, if you really want to derive from Gtk.Widget then you also have to set up a drawing background. Gtk.Misc does that for you, but Gtk.Widget could be a container that doesn't actually draw anything itself. But inquiring minds want to know, so you could do it like so:
from gi.repository import Gdk
class ManualWidget(Gtk.Widget):
    __gtype_name__ = 'ManualWidget'
    def __init__(self, *args, **kwds):
        # same as above
    def do_draw(self, cr):
        # same as above
    def do_realize(self):
        allocation = self.get_allocation()
        attr = Gdk.WindowAttr()
        attr.window_type = Gdk.WindowType.CHILD
        attr.x = allocation.x
        attr.y = allocation.y
        attr.width = allocation.width
        attr.height = allocation.height
        attr.visual = self.get_visual()
        attr.event_mask = self.get_events() | Gdk.EventMask.EXPOSURE_MASK
        WAT = Gdk.WindowAttributesType
        mask = WAT.X | WAT.Y | WAT.VISUAL
        window = Gdk.Window(self.get_parent_window(), attr, mask);
        self.set_window(window)
        self.register_window(window)
        self.set_realized(True)
        window.set_background_pattern(None)
Edit and to actually use it:
w = Gtk.Window()
w.add(SimpleWidget())
w.show_all()
w.present()
import signal    # enable Ctrl-C since there is no menu to quit
signal.signal(signal.SIGINT, signal.SIG_DFL)
Gtk.main()
Or, more fun, use it directly from the ipython3 REPL:
from IPython.lib.inputhook import enable_gtk3
enable_gtk3()
w = Gtk.Window()
w.add(SimpleWidget())
w.show_all()
w.present()
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