If contractions are allowed, then the following is the best I can come up with:
I'm I.
Edit: Oh yes, 'I' is not a proper name, it is an indexical. If it were a proper name, than its rerferent would be context invariant (i.e., my use of 'I' and your use of 'I' would refer to the same object). In this regard, 'I' is similar to words like 'now' and 'here', and dissimilar to words like 'A' (which I saw you outlawed by fiat earlier). The referents of 'now' and 'here' change depending on the context of utterance, hence they are not proper names, just like 'I'.