Actually, I think technically (I can't think of a case where this has happened, though, so it would depend on the length of time between the axels, etc. whether they chose to call the third axel as a new jumping pass)...but if the 3 axels did get deemed a sequence, it would be an invalid element, since you can't have 3 listed jumps in a sequence. It would therefore take up a box and get 0 points.
You can have unlimited number of listed jumps in the sequence, but you only get points for the two with the highest point value (times 0.8).
If they're all the same jump, then you'd just get credit for two of them.
As you say, what happens between the axels would matter. If there are steps or turns other than the step from back outside landing to forward outside takeoff, which is specifically allowed for axels at the end of sequences, then they'd be considered separate elements.
If you did axel-tap toe-axel-tap toe-axel, it should count as a sequence with points for two axels, times 0.8..
If you did single axel-tap toe-double axel-tap toe-triple axel, it should count as a sequence with points for the double and the triple, times 0.8. Waste of points -- if you can do a triple axel you'd be better off getting full credit for it.
However, it's not allowed to do more than two double axels in a program. So if you break that rule by doing a sequence of three double axels, and the whole sequence counts as one element, then the whole element won't count. Even bigger waste of points.
At lower levels, it may not be allowed to do more than two single axels, so the above would apply with singles.
There are lots of feats that can be done on the ice and that can be artistically exciting but that don't meet the demands of competition. This includes long sequences repeating the same element over and over again, it includes illegal moves like backflips and (for pairs) detroiters and headbanger spins. Those kinds of sequences or moves have always been saved for exhibitions/show skating.
Then there are other moves or ways of combining moves that were perfectly valid although generally unusual in 6.0 programs but that don't fit the current well-balanced program rules.
On the one hand, if skaters are going to earn points for elements there need to be some kind of guidelines on how many of each kind of element, and how many total elements, they're allowed to earn points for. Otherwise
Even under 6.0 since the mid-1990s the long program rules were modified to produce "well-balanced" programs that all had similar element content by encouraging or requiring minimums of certain kinds of elements and imposing maximums on other kinds.
On the other hand, the current rules are, IMO, unnecessarily specific about the limits on each kind of element and make it illegal or a waste of points to attempt certain skills or combinations of skills that could have been appropriately rewarded (or ignored -- we never knew what any judge thought about any of the specifics) when whole programs were judged instead of specific elements.
Still, as long as skating is a sport, we have to expect there to be some attempt at standardization so that everyone is competing on a level playing field. Not everything that works artistically can fit in the competitive format.