Consider the following streaming ``algorithms'':
- When an element $x$ arrives, flip a fair coin until it shows ``heads''.
- Fix a random hash function $h:U\to\{0,1\}^\infty$ (i.e., it maps elements into infinite bit strings). When an element $x$ arrives, do $LZ(h(x))$ operations, where $LZ$ is the number of leading zeros in $h(x)$. Notice that for any element $x$, the distribution of time spent is same as in (1).
Clearly, both algorithms spend in expectation $O(1)$ time per element, and their worst case is unbounded.
The key difference is that algorithm (1) makes the coin flips independently for each arrival, and (2) fixes a distribution such that every time $x$ arrives, it spends the same amount of time.
(1) should clearly be called amortized constant time. (2), however, might spend more than constant time if the input consists of many appearances of an element $x$ for which $LZ(h(x))$ is large.
Would you consider (2) to have an ``amortized'' constant time?
If not, is there anything more accurate than saying that it has "expected constant time per element"?
While these ``algorithms'' are clearly useless, there exist algorithms with a runtime similar to (2) (e.g., see here).