Strongly NP-complete variants of subset sum or partition problem
It's like cheating, but if you change the representation of the numbers in the input then the problem becomes strongly NPC:

SUBSET SUM OF FACTORIZED SEMIPRIMES "SUBSUMS"

Input: A list of $N+1$ integers: $q_0$ and $A = \{q_1, ..., q_N\}$ each one represented as a (sub)sum of factorized (semiprime) integers; i.e. $q_i$ is given as $p_{i,1} 2^{a_{i,1}} + ... + p_{i,m_i} 2^{a_{i,m_i}}$ where $p_{i,j} \geq 3$ is a prime and $a_{i,j}$ is its "shift" exponent (in other words the input is the list of integers $p_{1,1}, a_{1,1}, p_{1,2}, a_{1,2},  ..., p_{2,1}, a_{2,1}, p_{2,2}, a_{2,2}, ...$)  

Output: Does there exist a subset of $A' \subseteq A$  such that $\sum_{q \in A'} q = q_0 $?

Easy reduction from the strongly NPC problem $X3C$.