Can you use the -masu te form to form requests? Like, can you say "machimashite kudasai"? (O.o sounds weird huh.)
～ましてください is ungrammatical, because only one (in most cases, last) verb is allowed to take polite form per sentence to make that sentence polite. Since ください is already the polite form of くれ (くださる < くれる), no other polite verbs allowed.
Strictly speaking, a sentence that consists of multiple coordinate clauses allows (or is recommended) to have polite form for each clause, because they are but independent sentences glued together. て indeed sometimes works as coordinate conjunction when it leads a clause, which may cause some confusion, but I could hardly think of a valid setting ～ましてください would separate into two clause at て.
東京の大学に行ったが、故郷で就職した。 (only as coordinator)
東京の大学に（○ 行きました／× 行った）が、故郷で就職しました。
I went to college in Tokyo, but got a job in my hometown.
東京の大学に行くのに、とてもお金がかかる。 (only as subordinator)
東京の大学に（× 行きます／○ 行く）のに、とてもお金がかかります。
It costs much money to (=so that I) go to college in Tokyo.
東京にも（○ 行きまし／○ 行っ）て、シアトルにも行きました。
I went to Tokyo, as well as (=and I went to) Seattle.
東京に（× 行きまし／○ 行っ）て、人が（× 変わりまし／○ 変わっ）てしまいました。
He has become totally another man since he went to Tokyo.
Typically the "~mashite" form is only used for pretty specific cases, like phrases such as "hajimemashite" and sometimes in mid-sentence when speaking very politely.
But "~mashite kudasai" would be quite unnatural and I have never heard or seen this used before.
As rhyaeris has pointed out in a comment, I have an article about this, so feel free to check it out if you like.
As a general rule, I think it is best for learners of Japanese to stick to a single "~mashite" per sentence, unless they have become extremely comfortable with a certain expression or form.