I have to export my illustration as the best resolution
The best resolution is not a humungous overexaggerated resolution.
The best resolution is the one you need for the project.
So, the best resolution for a website could be 500px, or for a banner could be 6000px.
But if you want a humungous resolution to be printed at a really big format, leave the image as vectors. That is what vectors are for, they are resolution-free.
Or can you recommend any programs, which are able to export ai/eps/pdf files without limits?
I believe, "without limits" is limited to the physical properties of the known universe. So, not even a quantum computer or all the computer power of the humanity can.
File size is limited by either the file format, or the operating system, or the physical capability of the computer.
For example, a bitmap image could be "only" about 32,000 x 32,000 px.
On Windows, it could be for example 2 or 4Gb