Most auxiliary languages I know of, especially Esperanto, are primarily a posteriori—that is, their vocabulary or grammar are derived from other existing languages, and most are fairly Eurocentric. While many might have a priori grammar, but not vocabulary, I have yet to see another language that is entirely a priori—not derived from other languages. Do any such languages exist? Note I am talking about languages designed to be an international auxiliary language.
Short answer: I've never seen or read a description of what you're seeking.
Long answer: What you're seeking is a most unlikely beast, reason being...
Generally speaking, auxiliary languages seek being completely or nearly completely a posteriori for reasons of simple expedience. Common design goals of an auxiliary language are ease of learning and simplicity of use. These goals are best met when vocabulary and grammar are familiar to the target audience. This is why Interlingua and Esperanto are considered "easy" languages to learn and have gained considerable user communities, with Esperanto boasting a small community of native level speakers.
What you're describing, a completely a priori invented language is most usually the domain of the artistic language, the engineered language and the philosophical language. With the exception of the philosophical inventors, most glossopoets are not interested in propagandising their languages as auxiliary languages. Occasionally a philosophical language inventor will propose his language as a means of general communication, but this is usually done from a position of ignorance as to how language works (accomplished auxiliary language proponents are usually quite well aware how language works in the wild!).
Auxiliary languages tend not to be classified as a priori simply because they impose layers of difficulty that run contrary to the usual design parameters of the genre. However, it can potentially be argued that a priori invented languages can become auxiliary languages after a fashion and if the appropriate stars align. For example, it may be that Klingon has seen some use as an ad hoc auxiliary language if, for example, two groups of Star Trek fans who do not share a language in common other than Klingon happen to meet at a SciFi convention.
Needless to say, even if there were an actual auxiliary language that is classified as purely a priori, it certainly isn't famous. Otherwise, we'd know about it!
We will have to go back some centuries in time to find a language like this. Since the success of Esperanto (and, to a lesser degree Interlingua, Glosa, and Toki Pona) it seems to be clear that an auxiliary language has to be naturalistic in some way to catch on. Nowadays, a priory languages (e.g., Loglan and its forks) are only designed with other purposes.
With this in mind, probably Solresol (1817) is the most famous a priory auxiliary language.