All the elections I'm aware of use secret ballots, meaning that no one knows which candidate a different person has voted for. But surely there must be elections somewhere where that's not the case and each vote is public ("Open ballot system")? If so, what's the largest election to use non-secret ballots?
This is a somewhat tentative answer because voting secrecy is alas a matter of degree (FYI: as of 2015 at least, Sweden still had vote-organization rules that resemble those in Germany or the USA from more than 100 years ago, meaning party-identifiable ballots.)
Wikipedia suggests that the 1993 presidential election in Nigeria used a system that counts as "open"/non-secret. Actually, the gubernatorial & state-legislative elections in Nigeria in the previous years (1991-1992) undoubtedly count that way and that is so because the voters simply lined up in different queues depending on the candidate/party supported. The presidential vote that followed used a slightly modified system, but not by much:
The open ballot was first introduced in Nigeria in 1991 for the gubernatorial and state legislative elections. Section 87(2) of Decree No. 50 of 1991 expressly stipulated that "voting shall be by open ballot". Schedule 5 of the Decree gives an insight as to what an open ballot system means. It provides that at the close of accreditation, the presiding officer shall loudly announce the total number of accredited voters entitled to vote at polling station and explain the voting procedure to be followed. Furthermore, he is expected to introduce the candidates or their posters and symbols, the poll clerks and the polling agents. Thereafter, he has to call the roll of accredited voters and ensure that posters bearing photographs of candidates are well placed before the commencement of voting. Immediately the presiding officer announces the commencement of voting, accredited voters are requested to line up in a single line in front of the candidate of their choice. Separate queues are formed for men and women in areas of the country where the culture does not permit the mingling of men and women in the same queue. Once the queues are formed, security agents or poll orderlies are requested to stand at the end of the queue behind the last accredited voter. Voters are then requested to show their voters' cards duly stamped and the right thumb bearing an indelible mark made by the presiding officer. The presiding officer then loudly counts the number of accredited voters and then enters the votes scored by each candidate in a statement of result form. It is his duty to cross-check the total number of votes cast at the polling station with the total number of people accredited to vote at the polling station. The introduction of the open ballot system is not unconnected with allegations of massive rigging of past elections in Nigeria. To some, electoral malpractices and irregularities are traceable to the counting of votes which are cast in secret and which are open to abuse by functionaries of the NEC and politicians. In a predominantly illiterate setting, some have equally argued that the ballot box which is a symbol of secret balloting is totally strange.
The adoption of the open ballot system at the gubernatorial election was sufficient to expose the inadequacies of that pattern of voting in elections. It received a major dent at the National Assembly election which followed. It is therefore understandable why the NEC reviewed the system at the presidential election of 12 June, 1993 and adopted the open secret ballot system. This was only a slight improvement on the open ballot system. Though voters thumb- marked their cards in secret corners, they still voted in the full view of all present, so that it was still possible to know who a particular voter had voted for.
However, one has to keep in mind that that election was cancelled before results were officially announced, (and the likely/unofficial winner was also arrested) so the election was non-effective. Although its turnout was rather low, around 30%; in absolute numbers, about 14.3 million voted in that election, which is actually an important angle to this question. (The turnout for the 1991 election, which is even less debatable in terms of secrecy of the vote, was apparently even lower, around 15%.)
Even If you're willing to relax the notion to ballots that with some effort are discernible by a party or the state as to whom the voter voted for, the elections in Germany before 1903, or in the US before 1888, or those in communist Poland, or those in Sweden today had/have fewer participating voters than those 1993 Nigeria ones.
In the Swiss canton of Glarus, as of 2020¹, cantonal affairs are still decided by the Landsgemeinde. This is a general assembly that gathers every year since 1387. All men above age 16 (and since 1971 also women) who have active suffrage gather in the town square to decide on matters. All have the right to speak and vote. Since voting happens by raising hands, it is all public. Cantons in Switzerland have a lot of power, and among the decisions taken such have been to prohibit child labour (1856), introduce old age pension (1916), introduce women's suffrage (1971), and reduce voting age (2007).
The Canton of Glarus has a population of around 40,000 inhabitants. It is the largest entity that still uses a Landsgemeinde to take decisions. In smaller municipalities in Switzerland this form of direct democracy is still somewhat common, but usually the right is limited to voting and not everybody has the right to speak.
Source: Samuel Trümpy, Wikimedia Commons