I think you're misinterpreting what it means for a number to be uncomputable. It does not mean that the number is impossible to represent. It means that the number is impossible to identify the value of from its definition.

For example, consider the two thousandth Busy Beaver number. It's a finite integer by the nature of its definition, so if someone actually knew its value and had enough time, space, and materials it would obviously be possible to write it down. At the same time, it has been proven to be uncomputable. This is not a contradiction, because it just means that no one will ever knowwhichinteger is "the two thousandth Busy Beaver number".