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What does mari-mas mean? Typically said very politely and with a smile.

You need to give us a couple more syllables

"mari" is a part of the verb that they were saying
"masu" is a polite verb ending that can get tacked onto any verb in polite conversation

For example
Tomarimasu = I/you/he/we/etc. stop(s)
Komarimasu = I/you/he/etc. am/are/is in trouble
Atsumarimasu = we/they get together

and various other possibilities...

If you're requesting someone's services and explaining your request, you often hear: kashikomarimashta = I have understood.
(This would be renderred as "Yer all set" in Boston).

Could you have only heard part of the phrase? I'm thinking you might have heard ittemairimasu, which is a leave-taking greeting. Kind of like "so long" or "see you later," only more polite.

JR

Like my Brazilian colleague who would ride the elevator and the recorded voice would say ikkai-ni-mairimasu, or whatever floor, he would repeat it as best as he could. He spent a month in Japan, and he repeated it every time he got on on the elevator, and in the entire month I don't think he even came close to pronouncing it correctly.

Are you possibly thinking of "mairimasu," which means "I (humbly) come/go"?

This question reminds me of the time a Japanese client phoned while my mother was visiting. After I hung up from the rather lengthy conversation, my mother asked, "What does 'mash'ta' mean?"

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Usually the second 'a' is elongated. Context is usually a shop clerk who is saying a lot of words or as part of a subway announcement. Quite possible there are more syllables.

Well, as previously posted, masu (pronounced mahs - the trailing 'u' is almost unvoiced) is a polite suffix for many verbs. Since verbs are normally at the end of sentences, it could be any of a whole list of announcements. "Have a nice trip," "watch your step," "beware of the approaching train," "please come again," "we have bananas today," almost anything. Could have been mairimasu, arimasu, or any number of other sentence-ending verbs. Kind of like asking what "-king" at the end of a sentence in English means. Could be talking, walking, hiking, smoking, almost anything.

JR

My mother picked up on this on her first day in Japan. Particularly on the boat between Asakusa and the Intercontinental Tokyo Bay (just past Hama Rikyu).

Thanks to MrLapLap, we got a distilled translation of what the guide was saying (and she was excellent; friendly, great diction, very professional). My mother felt she was ending every other sentence with �arimas. So richarddd, you aren�t alone.

Thanks. That all makes sense.

I still wonder why so many clerks do long speeches in Japanese while ringing up my items, although it's obvious I don't have a clue what they are saying; and why so many adults have small doll or cartoon figures hanging from the their backpacks or other bags; and ...

If a clerk in England didn't say please and thank you to a customer who clearly didn't speak English, as a fellow customer standing behind them, I would be mildly angry on their behalf.
I very much appreciate that I'm treated as any other customer in Japan. It's surprising how much more you can pick out of the 'scripts' on each subsequent visit.

As for dolls and cartoon figures - almost everyone has one dangling from their phone (which makes for easy identification) and it's become a huge industry. The choice of mascots available is bewildering. Goodness knows how many billions of yen this whole business is worth. It seems natural to me that people will 'tag' their bags in the same way.

Remember, most Japanese love shopping. Most Japanese don�t have much room to store their purchases in/on. Therefore, things you can buy that don�t take up much room tend to be extremely popular.

In Europe, people buy tons of ornamental crap for their mantelpieces and shelves. It makes sense to have a figurine dangling from your bag instead.

That one is pretty simple. They almost always chant out what the item is and what the price is as they ring it into the register. They will also politely read back the currency tendered and amount of change as they give it. If you have several items, it can make for quite a little speech. In this country, we are so used to hearing "beep," bleep," "boop" as each item is passed over the scanner and only "there you go" when we get change that it seems strange. I like it.

JR

Long, long ago I had to deal with a young woman (probably fresh out of college) at a security checkpoint in Haneda airport, who said (in Japanese) "Please make sure that you remove all metal objects, phones, keys, coins and so on from your pockets before you walk through the metal detector... although there's no point me standing here and saying this since it's obvious that you don't have a clue what I am saying"

I complained. She was very embarrassed. Deep down I thought it was rather funny....

I still remember this very fondly on every visit I made to Japan. However, as I am hearing-impaired, I read their lips when they speak -- its a cheery feeling when they count your change out at the cash register in the big stores! (And yes if you ask, I don't count the cash again after I leave!)

Sanosuke!

Ho Ho!

I always imagined this would have to happen from time to time

I wonder if the OP isn't mishearing "narimasu", as in "Sen-en ni narimasu", or "That'll be 1000 yen".

Slightly OT but still related to narimasu, I was so relieved a few years ago to find out the usage "Kochira wa kohi ni narimasu" is not technically correct in Japanese. I always wanted to know when it would become coffee!