Dik (2003) gives an analysis of Greek nominative pronouns that suggests that there are occasion were even the nominative forms of ἐγώ and σύ maybe used as postpositive enclitics rather than full syntactic (and thus mobile) words. Her abstract:

The nominative forms ἐγώ and σύ of the Greek personal pronoun are traditionally described as exclusively ‘emphatic’. This paper argues, on the basis of examples from Plato and Sophocles, that these pronouns are often better understood as unemphatic, as becomes clear from their treatment as postpositives rather than ‘Mobile’ words.

Dik 2003

Her analysis is persuasive and substantiates statements in earlier studies on Greek pronouns and enclitics (among others: Marshall 1987).

For some time I have been curious about whether there are places in the Koine where this also occurs, but have not taken the time to consider the question. While working on a separate project on pronouns in the Septuagint, I came across the following example:

ἐγένετο δὲ ἡνίκα ἤγγισεν Αβραμ εἰσελθεῖν εἰς Αἴγυπτον, εἶπεν Αβραμ Σαρα τῇ γυναικὶ αὐτοῦ, Γινώσκω ἐγὼ ὅτι γυνὴ εὐπρόσωπος εἶ,  ἔσται οὖν ὡς ἂν ἴδωσίν σε οἱ Αἰγύπτιοι, ἐροῦσιν ὅτι Γυνὴ αὐτοῦ αὕτη, καὶ ἀποκτενοῦσίν με, σὲ δὲ περιποιήσονται
It happened when Abram came near to enter into Egypt, Abram said to Sarai, his wife, “I know that you are a beautiful woman. It will happen, therefore, when the Egyptians see you, they will say, ‘This is his wife.’ And they will kill me, but they will keep you. (Rahlfs & LES Lxx Gen 12:11-12).

Here we have ἐγὼ immediately following the first word in the prosodic unit, the verb, in a context with high probability that γινώσκω is receiving a particularly strong sentence accent. This seems like a strong candidate for interpreting ἐγὼ as postpositive.

A quick search suggests that there are others that share the pattern in non-translation Greek, but it is unusually common in the Shepherd of Hermas:

λέγω αὐτῷ· Κύριε, οὐ γινώσκω ἐγώ τί ἐστιν εἰς θάνατον καὶ τί εἰς καταφθοράν
I said to him, “Lord, I don’t understand what ‘to death’ means and what ‘to corruption’ means” (Hermas, Sim. VI, ii, 3).
καὶ ἐκύκλευον ἐγὼ μετʼ αὐτοῦ.
And I went around [the tower] with him (Hermas, Sim. IX, ix).ἔγνων ἐγὼ ἐκ τῶν προτέρων ὁραμάτων ὅτι ἡ Ἐκκλησία ἐστίν
I knew from the previous visions that she was the Church )Hermas, Vis. IV, ii, 2).
ἔδοξα ἐγὼ ὅτι πάρεστιν ἐκπειράζων με, καὶ λέγω αὐτῷ· Σὺ γὰρ τίς εἶ;
I thought that he had come to tempt me, and I said to him, “Well who are you? (Hermas, Vis. V, 3).
Λάβε, φησίν, καὶ ἀποδώσεις μοι. (4) ἔλαβον ἐγώ
“Take it,” she said “and return it to me.” I took it (Hermas, Vis. II, i, 3–4)

Further exploration of this sort of pronominal usage in the Koine period would be a worthwhile project for someone.