I assume that it was considered at one point, and that there's a good reason for not doing it, I'm just curious as to why. I'm sure that there are many people who struggle to remember a single password for a site, but aside from that, would requiring two or three passwords to login to a website really enhance security, or would you gain a similar benefit by just increasing your password complexity?
It depends what you mean by "enhance security". On the one hand, yes, it's more pieces of information for a thief to steal, but on the other hand, if they can steal one password, chances are they can steal all your passwords.
The Cole's Notes is: we like to have multiple authentication elements which
fall under different categories with respect to "something you have", "something you are", and "something you know".
Having authentication elements from more than one category (ex.: a password and a fingerprint, or a password and an SMS code on your phone) is properly called multi-factor authentication, while having multiple elements from the same category is called mutli-step authentication.
Stealing "something you know" typically requires planting a keylogger or file sniffer on your computer, watching you type your passwords, stealing the paper you wrote it down on, etc.
stealing "something you have" typically requires lifting something from your office after hours, stealing your laptop bag with the key gen token in it, etc
stealing "something you are" typically requires some kind of access to your biometrics.
So having multiple elements from the same category does add some barrier to theft, but not much since stealing all the passwords is not much harder than stealing one password, stealing all the physical objects is not much harder than stealing one physical objects, etc. Usually the inconvenience far outweighs the added security. It's much better to require thieves to perform a data theft and a physical theft, etc.
The main concern with implementing multiple passwords is that users tend to have a hard time remembering one password, and adding more just makes it more likely that they will start writing down each of them, netting you with a loss of security.
Besides, even if they don't write it down, there is still a single point of failure and multiple passwords will not protect you against the most common ways to steal them, being shouldersurfing, Man-in-the-Middle and a databreach.
Third, what is to stop a user from using Passw0rd, Pa$$word and P@ssword or something similar along those lines? They will be very inclined to use variations of the same password or use the same combination they used on a different website that would implement the same thing.
All in all passwords are not really as secure as most people think, and are mostly there as a hurdle for the common threats, but are in no way perfect. The implementation of the multiple-passwords-concept that does seem to increase the security is 2FA.
Having a double/triple password makes the login process more cumbersome for users and offers no additional security because once the client machine or the server is compromised, the attacker has access to all passwords; it doesn't make any difference if the user has one or two or three passwords to login.
What actually increases security is multi-factor authentication.
Note that a "secret question" in addition to a password is not a multi-factor authentication.