Random Riddle: Thing 1 and 2

Later is when I became known
As an infinite figure, when shown
You’ve counted my spaces
Over two billion places
And still, my amount is unknown

(Thing 2)
I, too, am not young
I’m almost as old as Thing 1
I’m just a frog
On the natural log
But I can make counting fun

(Thing 1 and Thing 2)
When you combine us two
In the order of Thing 1 and Thing 2
We’ll be a baked treat
That’s painful to beat
Whether cherry, peach, or aloo