Assume RSA with a public modulus $N$ of $n$ bits, a small odd public exponent $e$, plaintext $M$ a random non-negative integer less than $2^m$ for some integer parameter $m$, with $M\mapsto C=M^e\bmod N$ (textbook RSA) used for encryption. Inasmuch as it matters, assume there are $2^r$ ciphertexts available to the attacker, and her goal is to recover one plaintext with odds better than $2^{-k}$. If necessary also assume $n\ge1024$, $e\le2^{16}+1$, $r\le30$, $k\ge40$, or/and $M$ is of exactly (rather than at least) $m$ bits.
If $M^e<N$, then $M^e\bmod N=M^e$ and thus $M=C^{1/e}$, making decryption trivial by $e$th root extraction. $m\ge n/e+k+r$ makes us safe from trying that for each available ciphertext; but not necessarily against improvements, or other attacks.
Up to what bound for $m$ do we know an attack better than factoring $N$ using GNFS?
As an aside: starting with what bound for $m$ (if any) is there a positive security argument?
Updates (main question, now highlighted, remains without an answer proposing a bound):
As rightly pointed by D.W.'s in this answer, regardless of $e$, we must also have $m$ big enough to resist being found by "square-root" attack, a "meet-in-the-middle" search using a space/time tradeoff. To be safe from that we can make $m\ge a+b+k$ where the adversary is powerful enough to make $a$ accesses to a memory system of $b$ bits, say $m\ge 90+70+k$.
There is a simple extension to the $e$th root attack: the adversary hopes that an $M$ is divisible by some small integer $s$ of her choice, computes $C'=C\cdot s^{-e}\bmod N=(M/s)^e\bmod N$, then applies the $e$th root attack to $C'$. Perhaps she chooses $s$ as the successive primes, or some better method. This considerably extends the range of vulnerable $m$ (to, uh..).
Note: I assume all plaintext is random and independent, and the adversary has no access to a decryption oracle (even limited to revealing messages other than the originals and with at most $m$ bits).
Another simple extension to the $e$th root attack: the adversary hopes that $k=\lfloor M^e/N\rfloor$ is small enough to be enumerable, and tests if $C+k\cdot N$ is an $e$th root for small $k$. That's independent, but only marginally extends the range of vulnerable $m$, as far as I can tell.