In

Hardy, G. H.; Wright, E. M., An introduction to the theory of numbers., XVI + 403 p. Oxford, Clarendon Press (1938). ZBL64.0093.03.

this is denoted $U(x) := \operatorname{lcm}(1,2,\dots,x)$. Its asymptotics are found in Chapter XXII, which leads up to the proof of the prime number theorem.

Write $\psi(x) = \log U(x)$. Relevant facts (going from easier to harder):

Theorem 414: $A_1 x < \psi(x) < A_2 x$

Page 343: $\psi(x) > \frac{\log 2}{4}\;x$

Theorem 420: $\psi(x) \sim x$

More recent result:

Rosser, J. Barkley; Schoenfeld, Lowell, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6, 64-94 (1962). ZBL0122.05001.

\begin{align} \psi(x) & > 0.84\;x\quad\text{for } x \ge 101 \\ \psi(x) & < 1.038\;x\quad\text{for } x > 0 \end{align}

Another title that sounds promising:

Rosser, J. Barkley; Schoenfeld, Lowell, Sharper bounds for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$, Math. Comput. 29, 243-269 (1975). ZBL0295.10036.

As noted, these estimates are known, but possibly not "combinatorial".

In

Hardy, G. H.; Wright, E. M., An introduction to the theory of numbers., XVI + 403 p. Oxford, Clarendon Press (1938). ZBL64.0093.03.

this is denoted $U(x) := \operatorname{lcm}(1,2,\dots,x)$. Its asymptotics are found in Chapter XXII, which leads up to the proof of the prime number theorem.

Write $\psi(x) = \log U(x)$. Relevant facts (going from easier to harder):

Theorem 414: $A_1 x < \psi(x) < A_2 x$

Page 343: $\psi(x) > \frac{\log 2}{4}\;x$

Theorem 420: $\psi(x) \sim x$

A consequence of 420 would be: $$ \text{Let }0 < A < e < B. \text{Then }A^x < U(x) < B^x \text{ for large }x $$ More recent result:

Rosser, J. Barkley; Schoenfeld, Lowell, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6, 64-94 (1962). ZBL0122.05001.

They prove:
\begin{align} \psi(x) & > 0.84\;x\quad\text{for } x \ge 101 \\ \psi(x) & < 1.038\;x\quad\text{for } x > 0 \end{align}

Another title that sounds promising:

Rosser, J. Barkley; Schoenfeld, Lowell, Sharper bounds for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$ (http://dx.doi.org/10.2307/2005479), Math. Comput. 29, 243-269 (1975). ZBL0295.10036.