Very few countries (if any) limit the number of citizenships you can hold per se and there are certainly no international norm that would do so.
What prevents you from getting many are rules like “if you apply for another citizenship, you lose citizenship X” or “to be naturalized, you need to prove you gave up your previous citizenship(s)”. And these rules can have exceptions depending on personal circumstances (e.g. whether you grew up in the country), international treaties (e.g. the Strasbourg convention), whether your other nationality can be renounced at all or for for specific citizenships (e.g. Germany ‘tolerates’ dual citizenship when the other citizenship is that of another EU country, even in cases in which the general rules would strip you of your German citizenship).
Spain also has a rather original concept of “dormant nationality”. You don't lose your citizenship by naturalising to an iberoamerican country but you cannot enjoy its effects until you come back to reside in Spain. Would it count in your tally?
Furthermore, even countries that have restrictive rules for naturalisations often do not mind multiple or dual citizenships acquired by birth (although even that isn't a rule: you can be effectively forced to choose between citizenships at a certain age, lest you lose one of them).
All this matters because it means you can't just cite a number, it also depends on how you acquire the citizenships in question and in what order. For example, there must be thousands of people who hold both the Turkish and the German citizenships and German law has recently been changed to allow it in even more cases but if you are an adult and you do not presently have them, there is no legal way for you to gain and hold both at the same time.
At most, we could put a ceiling on the maximum number of concurrent citizenships that one can hold. If we start with the number of countries in the world, and discount those that explicitly reject dual citizenship like the DR Congo and perhaps those that make it extremely difficult like Japan or Austria, we might still end up with dozens of potentially ‘compatible’ nationalities.
Looking at the question from the other end, I personally know people who have three citizenships by birth, have heard about people with four or five and can think of plausible ways to have a few more. Beyond that, there are two main issues that will limit how many you can really have in practice.
For citizenships acquired at birth, the main issue are registration or residency requirements. In many cases, parents who have never resided in a country cannot pass that country's citizenship to their children or need to register them fairly quickly (cf. notions like British citizenship “otherwise than by descent”). So if your parents have several citizenships (by naturalisation or otherwise) at the time of your birth, you might very well be born with three or more citizenships but you won't always have all of your grandparents' citizenships and you might not be able to transmit all of yours to your own children.
If you aren't born with multiple citizenships, the main way you could acquire others is through naturalisation. This brings some additional complications as the law sometimes explicitly specify that naturalisation is contingent on a willingness to integrate and make a life in the country. In a country like that, if it looks like you always intended to live elsewhere, your naturalisation could theoretically be reversed.
Most countries also only offer naturalisation after 5-10 years of residence, require some knowledge of the local language and take a long time to process applications (and even then it's rarely automatic, naturalisation might very well be denied). While there is nothing preventing you from doing that in several places, as a practical matter you won't be able to get more than a few citizenships that way in your lifetime.
If we are cynical and look at the most convoluted scenarios, we might still add countries from which you can effectively “buy” a citizenship (e.g. through a large investment in a local company) and also “use” several marriages to gain a few more while waiting on naturalisation elsewhere but even that might take some time and it seems difficult to get anywhere near the “maximum”.
One useful resource if you want to dig deeper in all this is the UNHCR Refworld database, which includes an extensive collection of documents on citizenship / nationality law (but unfortunately you will have to read and understand them yourself to figure out the rules in each country, they don't provide a neat table or summary).
Finally, since you mentioned ease of travel in a comment, note that having several citizenships can help you benefit from visa waivers or allow you to take up residency in different places but if you are a citizen somewhere, it also means that this country does not have to give you access to consular assistance from your “other countries”.