I am currently playing the Turkish ( starting at 1205 - High period ).

The Mongols duly arrived ( around 1228 ) with several stacks ( 1st Horde ) appearing in Georgia, just as I was about to lay siege to it's "castle".
I opted to withdraw my forces to Trebizond and awaited the invasion of my little ( 6 province ) Turkish Empire.

[ NB : Other stacks in the 1st Horde also went to 2 ( or 3 ) provinces North of Georgia. ]

The Mongols decided to attack Armenia and I beat them off twice ( 2 separate invasions in successive turns ).
They then gave me a turn's respite ( during which a 2nd Horde appeared in Georgia )...
Most of the 2nd Horde moved off into "Russia" ( to join the 1st Horde ) leaving 3 stacks behind, two of which invaded Armenia again.
This time I drove them off the field of Battle ( even though I was outnumbered 2-1 ), as their General unit made a rash "lone" charge and got himself captured !

Can anyone out there tell me how many Horde "waves" will appear during the game ?

[ As you will see from my comments above, I have noted two Hordes to date. ]

NB : Each Horde arriving in Georgia had approx. 10-12 stacks. I presume further 2nd Horde stacks also arrived / appeared North of Georgia.