Vietnamese comes pretty close. Pronouns in conversation are almost all words for family members. The pronoun used depends on relative age. Speaking to a slightly older woman, for example, I would call myself em (little brother) and she would also call me em; she would call herself chị (big sister) and I would call her chị. So in a typical conversation there is no surface representation of person at all. There is no inflection so no question of verb agreement. I have even been in the situation of me and a Vietnamese speaker both calling each other and ourselves anh (big brother) when the age difference was not yet established.
The English translators of the highly amusing novel Dumb Luck by Vũ Trọng Phụng try to get the effect of this across by having characters speak to their family members using third person kinship terms instead of personal pronouns. I'm not sure whether it seems like that to Vietnamese speakers.
The only exception in ordinary circumstances is tôi / bạn. These are usually translated as I / you, but it's not quite as simple as that. Bạn means 'friend' and it is still used as a noun, so its second person meaning is derived from its primary meaning. I think someone told me that tôi originally meant 'servant' but I can't find any reference to that. Anyway, at a practical level, tôi / bạn do mean first and second person, but in my experience they are rarely used (I haven't been in many extremely formal situations). In most situations there is no need to use tôi / bạn for politeness as the age-relative pronouns express adequate respect. I have used tôi to address a mixed-age group; on the other hand, I can also speak as if addressing the oldest person by using the pronoun that expresses my age relative to that person's.
There are some more obscure pronouns or pronoun-like terms which are used in certain friendship and family situations and do have first- and second-person meanings (whether primary or derived I don't know). My wife has explained them to me but they are quite opaque to foreigners, who are at any rate unlikely to hear or use them. Wikipedia lists mình as a true first person pronoun but I can't agree. I think it's a reflexive with a derived first person meaning -- as in 'me, (my)self and I' -- and I have definitely heard it used as a second person reflexive, for example in an instruction in a meditation lesson: cơ thể của mình 'body belonging to self', i.e. 'your (own) body'.
I know the OP was about first person, but there is further evidence of the minimal appearance of person in Vietnamese in that the third person is represented only by the demonstrative ấy as in anh ấy 'he', literally 'that big brother'. On the other hand, there are words for 'it' and 'they'. Make of that what you will.