User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-24920136-20160725013617/@comment-24920136-20160728203911

the hereditary base n form actually is well defined if we copy the definition from the link you gave it is the form goodstein sequences use except with base 10 instead of 2, i believe it is a well defined unique representation for every n, however this still doesn't help OP escape the  e_0 trap