The smallest untweetable number

Which numbers can be expressed as a string of length $n$ on the alphabet $\{0,1,2,3,4,5,6,7,8,9,+,-,*,/,\wedge \}$? The numerals and symbols have their usual meaning, and $0\wedge 0$ is $1$.

Specifically, let $f(n)$ be the smallest natural number that cannot be so represented. Trivially, $f(0)=0$, $f(1)=10$, $f(2)=100$. With $n=3$, operations come into play but still $f(3)=1000$. With $n=4$, we have "$10\wedge 4$", and $f(n)=10^4+1$. A quick program gives $f(5)=10^5+1$. The obvious code for $n=6$ is inadequate, as expressions like "$9\wedge 9\wedge99$" are too large. Working by hand, I have convinced myself that $f(6)=(10^6+9)+1$. It is plausible that $f(n)$ will always be a sum of distinct powers of $10$, but I fear that there will be some perfect high power not much larger than a power of $10$ that let's one reach a bit farther. And for large $n$, recursive base 10 representations come into play.

This problem is inspired by Berry's Paradox: ``the smallest number that cannot be defined in at most 280 characters'' is a definition that contains at most 280 characters. See this substack post by Joel Hamkins for a more thorough discussion of Berry's Paradox.

The trivial bounds are $10^n \le f(n) \le 15^n$ for $n\ge 1$. An easy recurrence gives $f(n) \ge 10^n+f(n-(L(n)+4))$, where $L(n)$ is the number of characters needed to express $n$.

I ask for

1. a better upper bound, and
2. for more exact values, and
3. a good way to express the solution to the "easy recurrence", and
4. are there any natural numbers that are representable, but only using division or subtraction?