Exploring fun and horrible ways in which nondeterministic automata (NFA) determinization can go wrong. We consider ways to make automata, search for a worst case, and then make inductive leaps. All without the sanitychecking of a textbook or good research. What could possibly go wrong?
Automata Construction Possibilities
Consider the possible ways to construct 2state NFA for an alphabet = {0,1}.
 State 0 can accept or not. (2 options)
 State 1 can accept or not. (2 options)
 State 0 can transition, on ‘0’, to S0, S1, S0 & S1, or nowhere. (4 options)
 State 0 can transition, on ‘1’, to S0, S1, S0 & S1, or nowhere. (4 options)
 State 1 can transition, on ‘0’, to S0, S1, S0 & S1, or nowhere. (4 options)
 State 1 can transition, on ‘1’, to S0, S1, S0 & S1, or nowhere. (4 options)
Next, consider the same using 3states.
 State 0 can accept or not. (2 options)
 State 1 can accept or not. (2 options)
 State 2 can accept or not. (2 options)
 State 0 can transition, on ‘0’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
 State 0 can transition, on ‘1’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
 State 1 can transition, on ‘0’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
 State 1 can transition, on ‘1’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
 State 2 can transition, on ‘0’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
 State 2 can transition, on ‘1’, to S0, S1, S2, S0 & S1, S0 & S2, S1 & S2, S0 & S1 & S2, or nowhere. (8 options)
Though repetetive, exhausting the possibilities for both 2 and 3state nondeterministic finite automata exposes a trend. The amount of NFAs we can form using those states, #PossibleNFAs = AcceptPossibilities * TransitionPossibilities, where AcceptPossibilities = 2^^{#states} and TransitionPossibilities = (2^^{#states})^^{(alphabetSize*#states)}.
Some time later, I’ll brute force the space for a 2letter alphabet on 3 states, but for now, here my approximation of the worstcase:
1 2 3 4 5 6 7 8 9 

Before determinization, it looks like this:
A little transitionheavy, but seemingly unassuming. If you turned it 90° clockwise, it would make a convincing muppet. Anyway, after determinizing it using the subset construction (and not using any minimization techniques) it looks like this:
Pretty big.
This post was primarily a formatting test. Expect more soon.