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We have following sequence:

12345, 33552, 45624, 66345, 73557, 45678, 66885, 78957, 99678

What will be the next number?

Answer is known now (@JonMark Perry and @Mark found it). The answer is:

Danger! It is the correct answer:

106890

or

A688A

but they found it by some properties of this sequence not by general rule and this general rule is very different (And it is interesting that this sequence has so many properties!).

Now the question is: what is the general rule?

Hint 1:

notice that always two of five digits are the same (but in different place) like in the previous one number.

Hint 2:

There are always the same two digits

Hint 3:

For example 12345 -> 33552. Next 33552 -> 45624 and so on.

Hint 4:

What with other digits?

Hint 5

Because of general rule, the answer (10th number in sequence) is the first one that doesn't pass rule 99678 -> 106890 But it can be fixed if we choose A688A answer.

Not sure, but:

$A688A$, or $1068810$ if not using single digits

because:

the sequence repeats every five, with (plus 3) mapped across all the digits. You can use the map from $66345 to 73557$ to map from $99678$.

Also:

the sum of digits divided by $3$ is $n+4$ for the $n^th$ term.

The general rule is:

$abcde to (b+1)c(d+1)e(a+1)$

The initial guess was that the answer is either:

106890, or 6880, or 6890, or 06880

Reasoning:

Digits of numbers no. 6 through 9 are numbers 1-4 plus 3. (12345 +3 = 45678 etc.). The 5th number is 73557, but its unclear whether the author intended to treat it as a whole number or separate digits. Therefore, I suggested more than one answer.

After the hint from the author:

The difference between numbers 1 and 2 is 33552-12345=21207, between 2 and 3: 45624-33552=12072, between 3 and 4: 66345-45624=20721. The difference shifts numbers, therefore the next one is 07212 (7212).

Checking it: 66345+7212=73557 (5th number). Continuing on, 73557+72120=145677; 6th number is 45678, so I guess the spare digit is added to the end. Moving on, 9th number is 99678 plus 7212 = 106890.

The final-final answer is:

Well, back to the very start... 106890.

P.S:

But the "regular math" is broken on 6th number anyways, as far as I'm concerned. :)

P.P.S:

Thanks @JonMark Perry for the edits. :)