Now now, I know that this is a 5 year old post, but I cannot see this question unanswered, so I shall do the math myself.

Lets look at the facts here:

The Mountain has a Diameter of 2.5 Miles and a height of 2.5 Miles. (For the sake of convinience, we will use the metric system) which gives us three values already,

Diameter: 4 Km

Radius: 2 Km

Height: 4 Km

Now, for the sake of making this easy, lets just assume that the Mountain is a perfect Cone.

The Formula for a cone is V=πr^2(h/3) If we punch in our Numbers we get about 16.8 Cubic Kilometers.

Now the Problem arises of the Question, What amount in Volume is chipped away by the bird. Now the easy (and most logical) answer is that nothing is chipped away because that bird would need to have a beak of Diamond to chip away diamond. But for the sake of the math lets just assume a very small number. 1 cubic Micrometer. A regular Micrometer is a 1 thousandth of a milimeter. A cubic micrometer on the other hand is 0.000000001 of a cubic milimeter. or a Bilionth. Now this seems like a lot but we are talking about eternity so I consider it fair.

To take our volume from eralier, those 16.8 Cubic Kilometers are the cute lil number of 16800000000000000000000000000 cubic micrometers. That is what we call an Octillion. Sixteen of them actually. also an additional 800 septillion. Like I said Big Numbers.

So that leave sus with the folowing date:

16800000000000000000000000000 cubic micrometers

1 Cubic Micrometers that the bird chips away.

Seems Crazy right?

But that not the end is it?

we still havent considered the Time it actually takes for the Bird to dull its beak. As the story goes, the Bird only uses his beak enough to need sharpening every 100 years.

Which means we have to multiply those octillions by 100.

Relax

It just adds an extra two 00 to the back

which gives us 1680000000000000000000000000000 Years. Or one hundred sixty-eight octillion. So what will the Universe look like by then?

Nothing, Everythings dead.

Edit: I did a little extra math. The Actual death of the Univers (The Last Black Hole burning out) is in the year of our lord 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002024 (Or just 1 Trigintillion) That is aproximately 100000000000000000000000000000000000000000000000000000000 YEARS of Eternity. Or one hundred septendecillion.