Cannot make document an exact size no matter what.

@serrinido those dimensions include bleed as well? With that image of yours, you require a bleed zone for it to print.

Digital images don't have physical dimensions, they only have pixel dimensions.

Pixel dimensions divided by PPI (pixels per inch) = Printed dimensions in inches.

Photoshop does the math for you, and might give you the impression that it's working with inches, but behind the scenes, it's all about pixels.

I assume that the PPI is set to 300.

You want 8.625 inches, multiplied by 300 = 2587.5 pixels.

It's not possible to have fractions of pixels, they have to be whole. So Photoshop rounds this up to 2588 pixels, which multiplied divided by 300  becomes 8.6266666 inches.

A few things could be going on here, some I'm not sure how to answer, but it's best to check off some possibilities:

1. If your image does not precisely line up pixel-perfect to that 8.625x8.75, and you're scaling using Image Size with proportionate scaling on, it will not give you the correct results. Has that been double checked?

2. What is the resolution of the image? If it's an unusual number of pixels per inch, and the .625 doesn't line up with a whole pixel amount, it might be that Photoshop is cropping the image to the pixel edge. Remember that Photoshop works in pixels, and there cannot be a fraction of a pixel.

3. If you create a new file with the size and resolution you want, and pull your desired image into there and save, does that work?

4. If the image is high resolution enough that you can afford to crop in a couple pixels, what happens if you make the Image Size just barely larger than you need (e.g., 8.76 inches tall), then go into Canvas Size and crop it to the size you want? Does that work?

I do agree that the .002 is probably not enough to cause any visible issue on printing, but I understand how frustrating this is. I have, seen a similar this issue crop up before when resizing my own images. Usually it adds a random .001 inches to the end with no real cause I can find, but that has never reflected in the finished product.

Photoshop is rounding to either 8.627 or 8.623 - not sure if its a bug or a limitation of the thousandth decimal point. Either way, how are you supplying these images to the printer? My assumption is if its .002 off the pages will just be cropped and its not going to affect printing. These dimensions seem to include bleeds so the pages will be trimmed anyway.