Thank you for your reply, Elle. It’s a fun problem. This question was asked of me by one of my students and I had to admit, that I had never really thought about it. So, last week I spent several days researching what the scholarly community thought. What I found I’ll hold off till later 
The NIV translation of this verse is misleading. I’ll get a bit technical here, but quick glance at most any Hebrew text book will substantiate what’s to follow.
But first before we dive into the weeds, most commercial Bibles to which I have access also use “greatly increase” or “greatly multiply” (e.g., RSV, NRS,NAS, NAU, NIRV, KJV, among others, including the English translation of the Septuagint). The problem with the other translations (you correctly cite the NIV’s translation, but the NLT gets it wrong as well) is that their translation ignores the underlying Hebrew grammar. Specifically the first word of the Hebrew phrase in question, harbah `arbeh, is an emphatic infinitive (see NOTES 1 and 2 below) and the second word is the verb phrase “I will increase”. Translated literally, the two words are read as “to be many I will make many”. Now, since an emphatic infinitive describes the following verb (i.e., “I will increase”) the translation is “greatly I will increase/multiply”.
Bottom line? The translation of “greatly increase” or “greatly multiply” is as close to accurate as English can get.
So, we’re still left with wondering to whom or what was the author comparing Eve’s soon to be painful pregnancy?
Blessings,
Michael
NOTE 1: Young’s Literal Translation of this verse reads, “Multiplying I multiply”. Young’s is on the right track, but is not quite there in that the translation doesn’t use an adverb. OTOH, this Bible dates back to the mid 1800s when the knowledge of BH was very, uh, superficial.
NOTE 2: An emphatic infinitive is an infinitive that occurs immediately prior to the infinitive’s verb. This form occurs frequently in the 2nd creation story, to die, you will die or to eat you may eat. Throughout the Hebrew Bible when you run across the phrase “surely you will…” or “certainly you will …”, you can bet the underlying grammar is an emphatic infinitive.