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There are at least $\ 2^{n-1}\ $ triangulations of $\ [0;1]^n\ $ into nice $n$-simplexes. I am quite sure that there are exactly $\ 2^{n-1}\ $ of them.

CONSTRUCTION   First I'll present one nice triangulation of $\ [0;1]^n$.

NOTE   Włodzimerz Kuperberg and myself have obtained this nice triangulation (see below; but of course we didn't use term nice) independently. I did it during the first half of 1996 (I don't know Włodek K's exact date). Looking back, it is closely related to the old homological/combinatorial triangulation of a prism (one may check for example a Pontryagin's small monograph on Combinatorial Topology; see Лев Семёнович Понтря́гин); small but great.

Let $\ \pi:\{1 \ldots n\}\rightarrow \{1 \ldots n \}\ $ be an arbitrary permutation. Let

$$\Delta_{\pi}\ :=\ \{(x_1\ldots x_n)\in [0;1]^n\ :\ \forall_{k=2}^n\ x_{\pi(k-1)}\le x_{\pi(x_k)}\} $$

The family of $n$-simplexes $\ \Delta_{\pi} : \pi\in S_n,\ $ together with their simplicial faces, forms a nice triangulation (which has $\ n!\ $ simplexes of dimension $\ n.\ $ We get the $\ 2^{n-1}\ $ such triangulations due to the $\ \mathbb Z_2^n\ $ action of isometries on $\ [0;1]^n$.

I feel that there are no other nice triangulations in $\ [0;1]^n$.

COUNTEREXAMPLE (Yes!) in dimension 3   If you see a counterexample in dimension $\ 3\ $ then here is a place to write down a $5$-th nice triangulation (just replace each star by a real number)--here simplexes are specified by their vertices:

1. simplex:   $((0\ 0\ 0)\ \ (1\ 0\ 0)\ \ (1\ 1\ 0)\ \ (1\ 0\ 1))$
2. simplex:   $((0\ 0\ 0)\ \ (1\ 1\ 1)\ \ (1\ 0\ 1)\ \ (1\ 1\ 0))$
3. simplex:   $((0\ 0\ 0)\ \ (0\ 1\ 0)\ \ (1\ 1\ 0)\ \ (0\ 1\ 1))$
4. simplex:   $((0\ 0\ 0)\ \ (1\ 1\ 1)\ \ (0\ 1\ 1)\ \ (1\ 1\ 0))$
5. simplex:   $((0\ 0\ 0)\ \ (0\ 0\ 1)\ \ (1\ 0\ 1)\ \ (0\ 1\ 1))$
6. simplex:   $((0\ 0\ 1)\ \ (1\ 1\ 1)\ \ (0\ 1\ 1)\ \ (1\ 0\ 1))$

Of course a picture can be nicer and (almost :-) as good.

@GjergjiZaimi -- thank you for your Question. Then, The Masked Avenge, David Eppstein and Pietro Majer--thank you for your solutions and comments; and TMA and David--for your patient dealing with my difficulties.