The wiktionary tells me for the individual elements that "μή is the negative of thought or wish"

Pretty much. It's one of the two common negative elements in Ancient Greek; οὐ(κ) is generally used with indicative verbs, while μή is used with subjunctives and optatives.

and that ει is a kind of relative conjunction

Technically true, but a bit misleading. The more relevant translation for your purposes is simply "if".

In other words, εἰ μή means literally "if not" or "unless". This combination can also be translated as "except" without really changing the meaning:

οὐ γὰρ ἔστιν ἐξουσία εἰ μὴ ὑπὸ θεοῦ
For no power exists unless [it comes] from God
For no power exists except [that which comes] from God

P.S. Following your tangent a bit, for the following word, some manuscripts say ὑπὸ, others say ἀπὸ. In this context, I like ὑπὸ more; both words fundamentally mean "from", but ὑπὸ can also mean "under the authority of", which seems right here. (On the other hand, ἀπὸ can mean "originating from", specifying the fundamental source, which also makes sense.)

P.P.S. Be careful with your transcriptions! "Unless" is εἰ μή, with an eta, while "I am" is εἰμί, with a iota. The two letters are pronounced the same in Modern Greek, but were very different in ancient times, as we can see from transcriptions into other languages.

The wiktionary tells me for the individual elements that "μή is the negative of thought or wish"

Pretty much. It's one of the two common negative elements in Ancient Greek; οὐ(κ) is generally used with indicative verbs, while μή is used with subjunctives and optatives. Μή is also extremely common in conditionals, which will imminently be relevant.

and that ει is a kind of relative conjunction

Technically true, but a bit misleading. The more relevant translation for your purposes is simply "if".

In other words, εἰ μή means literally "if not" or "unless". This combination can also be translated as "except" without really changing the meaning:

οὐ γὰρ ἔστιν ἐξουσία εἰ μὴ ὑπὸ θεοῦ
For no power exists unless [it comes] from God
For no power exists except [that which comes] from God

P.S. Following your tangent a bit, for the following word, some manuscripts say ὑπὸ, others say ἀπὸ. In this context, I like ὑπὸ more; both words fundamentally mean "from", but ὑπὸ can also mean "under the authority of", which seems right here. (On the other hand, ἀπὸ can mean "originating from", specifying the fundamental source, which also makes sense.)

P.P.S. Be careful with your transcriptions! "Unless" is εἰ μή, with an eta, while "I am" is εἰμί, with a iota. The two letters are pronounced the same in Modern Greek, but were very different in ancient times, as we can see from transcriptions into other languages.

The wiktionary tells me for the individual elements that "μή is the negative of thought or wish"

Pretty much. It's one of the two common negative elements in Ancient Greek; οὐ(κ) is generally used with indicative verbs, while μή is used with subjunctives and optatives.

and that ει is a kind of relative conjunction

Technically true, but a bit misleading. The more relevant translation for your purposes is simply "if".

In other words, εἰ μή means literally "if not" or "unless". This combination can also be translated as "except" without really changing the meaning:

οὐ γὰρ ἔστιν ἐξουσία εἰ μὴ ὑπὸ θεοῦ
For no power exists unless [it comes] from God
For no power exists except [that which comes] from God

P.S. Following your tangent a bit, for the following word, some manuscripts say ὑπὸ, others say ἀπὸ. In this context, I like ὑπὸ more; both words fundamentally mean "from", but ὑπὸ can also mean "under the authority of", which seems right here. (On the other hand, ἀπὸ can mean "originating from", specifying the fundamental source, which also makes sense.)

P.P.S. Be careful with your transcriptions! The negative is μή, with an eta, while "I am" is εἰμί, with a iota. The two letters are pronounced the same in Modern Greek, but were very different in ancient times, as we can see from transcriptions into other languages.

The wiktionary tells me for the individual elements that "μή is the negative of thought or wish"

Pretty much. It's one of the two common negative elements in Ancient Greek; οὐ(κ) is generally used with indicative verbs, while μή is used with subjunctives and optatives.

and that ει is a kind of relative conjunction

Technically true, but a bit misleading. The more relevant translation for your purposes is simply "if".

In other words, εἰ μή means literally "if not" or "unless". This combination can also be translated as "except" without really changing the meaning:

οὐ γὰρ ἔστιν ἐξουσία εἰ μὴ ὑπὸ θεοῦ
For no power exists unless [it comes] from God
For no power exists except [that which comes] from God

P.S. Following your tangent a bit, for the following word, some manuscripts say ὑπὸ, others say ἀπὸ. In this context, I like ὑπὸ more; both words fundamentally mean "from", but ὑπὸ can also mean "under the authority of", which seems right here. (On the other hand, ἀπὸ can mean "originating from", specifying the fundamental source, which also makes sense.)

P.P.S. Be careful with your transcriptions! "Unless" is εἰ μή, with an eta, while "I am" is εἰμί, with a iota. The two letters are pronounced the same in Modern Greek, but were very different in ancient times, as we can see from transcriptions into other languages.