I don't think it is possible to get the card with a dog between McGrath and Rin Tin Tin. When you get enough cards, the quantities of horses on the bottom row overflow and show as 65000+/2 (e.g. 65461/2, 65515/2, 65527/2). I think I've been playing this game too much!
Edit: 1 day later got the missing card "Pompey"
When the bottom row goes over 2 and you click on the card, the extra spills over onto the next page (e.g. extra bottom-row dogs give more cat cards), but there must be something broken there. I've finally got time to look into the reported bug...
And yeah, to get that many cards is a crazy number of games played!