Gaius, who is host to me and to the whole church, greets you.Richard Last, however, translates ξένος as "guest" instead of "host". See pages 62-71, most of which are available on Google books here. The Pauline Church and the Corinthian Ekklesia: Creco-Roman Associations in Comparative Context (SNTSMS 164; Cambridge: CUP, 2015). He finds that ξένος does not mean "host" in texts concerning associations, and this argument is not without weight. However, his theory creates more problems than it solves:
1. The name "Gaius" is a Latin praenomen (first name), and such names were reserved almost exclusively for intimate friends and family members. It is unlikely that a guest would be a close friend of either Paul or the members of the church of Rome. If Gaius was a guest we would expect Paul to use his cognomen rather than his praenomen.
2. Gaius was not a guest because he was already baptized (1 Cor 1:14). Richard Last counters that 1 Cor 1:14 might refer to a different Gaius:
Steven Friesen provides reason to think that this is not the same Gaius as in 1 Cor 1:14: he observes that the supposed household of Gaius, the ξένος (Rom 16:23), is never mentioned in the Corinthian letter and that 'it is odd that Paul said he baptized Stephanas' whole house (1 Cor 1.16) but he did not say he baptized Gaius's whole house (1 Cor 1.14).' Moreover, I would add, it is peculiar that Paul never commends Gaius's service as a host in the Corinthian correspondence, where he praises other service providers (1 Cor 3:1; 16:15-18).These are valuable observations, but they do not show that we are looking at two Gaiuses. They simply illustrate that "Gaius" was merely Stephanas's praenomen. Indeed, Last himself perceptively suggests that "Stephanas and other Corinthian service providers were crowned or honoured with inscriptions or proclamations" (p158), so he came close to realizing that Gaius had been named "Stephanas", meaning "crowned".
3. The believers in Rome would find it odd that a mere guest would choose to send greetings to them. Then, as now, people sent greetings to those with whom they had a strong connection. All, or nearly all, of the other greeters in Paul's letters had travelled on church business. A guest would not fit that pattern.
4. Paul would have little motive for sending greetings from a guest. Last proposes that Paul mentioned a guest to show the church of Rome that he (Paul) was able to bring in money by recruiting a fee-paying guest. However, for this to be plausible, Last would need to show that guests at all (or nearly all) association events actually subsidized the association. As far as I can tell, the examples that Last cites do not show that the guests paid for more than the food and wine that they consumed. Last also proposes that Paul mentioned the guest to show his addressees that he was able to recruit new members. However, if Paul wanted to display his ability to recruit, he would have mentioned actual converts, rather than a guest, who was merely a potential future convert.
5. The greeters in Rom 16:21-23 seem to be mentioned in a deliberate order. Paul gives priority to those who have been in the faith for longest and/or have traveled most widely for the gospel. Gaius is mentioned ahead of Erastus, who was already a believer (Acts 19:22), so Gaius was already a believer.
For these reasons I think we can be confident that Gaius was the host, not a guest.