Numbering items

If I want to name the quantity of times of something I talk about, I can do the following:

• I did it one time.
• I did it two times.
• I did it three times.

Analogue to this, I can use the following words:

• I did it once.
• I did it twice.
• I did it thrice.

Now, is there a name for the second example? Is there a way to call them, so that people know that I am talking about the workds once, twice, thrice?

Also, does it continue after thrice? Is there a way to say 4, 5, 6, ...?

Wikipedia ( don't laugh) refers to this class of number as multiplicative numbers Wikipedia. A similar question as this on Stack Exchange just over two years ago gave the answer as Numeral Adverbs,

In English, we only have the three words, once, twice, thrice. Thrice is rarely used these days, with 'three times' being more common. For numeral adverbs above three we use the appropriate numeral plus 'times', e.g. four times, five times , etc. In some languages other than English there is a mechanism that allows this class of number to go on indefinitely; refer to the link for Number Adverbs above for an example of one such language.

The first part to your question has an answer in this English Language & Usage Stack Exchange site post, which quotes William Fowler's English Grammar: The English Language in Its Elements and Forms:

§ 193. NUMERALS express the relation of Number and Quantity. They are divided into the following classes:

...

IV. NUMERAL ADVERBS, which answer the question How often? as, Once, twice, thrice, four times.

The second part to your question (does the series continue past thrice) is answered in this other English Language & Usage Stack Exchange site post:

Not according to the Oxford dictionaries:

These three are the only words of their type, and no further terms in the series have ever existed.