Triangles within triangles

Take any mention of triangle referred to in this solution to mean "equilateral triangle".

You can add any odd number, $2n+1 (n>0)$ of triangles to any existing triangle to make a new triangle by picking an edge and putting $n$ triangles along that edge and then $n+1$ more to fill in that edge for the new triangle. Here is an example for $n=3$ ( $7$ new red triangles added to the bottom edge of the big green one):

Now lets evaluate the possibilities for $N$ in the original problem.

So, starting with $1$ triangle, you can continue to add $3$ to it, to get these possibilities: $N = 1, 4, 7, 10, 13, ...$

You can add $5$ triangles to $1$ triangle to get $N=6$ , and $7$ to $1$ to get $N=8$ .

Now that you have a sequence of three possible values for $N$ (namely $6,7$ and $8$ ) you can make any higher number by adding $3$ to the lowest of the three, in this case $6$ , giving you $9$ , producing a new sequence of three ( $7,8$ and $9$ ). Rinse wash and repeat! :)

The only numbers that can't be constructed in this way, then, are $2, 3$ , and $5$ , the largest of which is $\boxed{5}$

And after playing around with these three numbers I convinced myself (without a rigorous proof... :-/ ) that there was no way to construct an equilateral triangle with any of these numbers of equilateral triangles.