How can I prevent LibreOffice Calc from removing the plus (+) sign?

Simply add a apostrophe when entering your data - turns the cell into a text cell :)

'+888888

If you need to use the number and cannot have it as text, you need to create a number format for it.

Right click and select Format Cells... -> Numbers. Enter your format into the Format Code field. The easiest format would be +0, which will display your number with a + and keep it a number, so it can be used in formulas.

Use +0;-0;0 as a number format

This answer automates the desired formatting for both positive and negative numbers.

Semicolons are required as separators between positive, negative and zero number formats.

The official LibreOffice documentation could benefit from a better example like this case.