Can you give me a big number with many divisors?

However, I am wondering whether there is a big(er) number, with a high ratio of divisors? Exists any number that e.g. has again a ratio of 2/3?

Yes, $\tau(6)/6 = \frac{2}{3}$

Note that in the best case scenario, $n$ has divisors $1,2,...,\sqrt{n}$ together with their co-factors $\frac{n}{1},\frac{n}{2},...\frac{n}{\sqrt(n)}$, so that's at most $2*\sqrt{n}$ divisors. This should allow you to figure out the upper bound to any number with a certain ratio. (and this also tells you that the ratio, while it may go up or down from number to number, will in general go down)

For example, for a ratio of $\frac{1}{2}$, you have an upper bound of 16, since for $n > 16$, $n$ has less than $n/2$ divisors.

If you try a few numbers, you thus see that 12 will be the last one with a ratio of $\frac{1}{2}$ or more

If $n = \prod p_i^{m_i}$ then $\tau (n) = \prod (m_i + 1)$

So we want $\frac {\tau(n)}{n} = \frac{\prod(m_i + 1)}{\prod p_i^{m_i}}$

The actual values of the prime factors are irrelevant to the number of divisors so to get a high ratio we want the primes to be as small as possible. So $p_i$ should be the $i$-th prime. $p_1=2;p_2 = 3;p_3 = 5; etc.$.

Which $p_i$ are raised to which power $m_i$ is irrelevent so to get a high ratio we want the high powers to go to the lowest primes so $m_1 \ge m_2 \ge m_3 ....$.

And obviously as we take higher powers $m_i$ the ratio of $\frac {m_i + 1}{p_i^{m_i}}$ is going to decrease radically.

That's not very precise but it gives a good frame work. For $n = p^m$ the highest ratio will be $n=2^m$ and the ratio is $(m+1)/2^m$ so for $n = 2,4,8,16... $ we have the ratio is $2/2=1,3/4,4/8 = 1/2,5/16, 6/32...$ etc.

For $n = p^mq^k$ the highest ratio will be $n = 2^{m+a}3^m$ and the ratio is $m(m+a)/2^{m+a}3^m$. So for $n = 6;12; 36;24;72;216 etc$; the ratio is $\frac 46;\frac 6{12};\frac 9{36}$ etc.

For no having three prime factors we have $n = 2*3*5; 2^2*3*5;2^2*3^2*5;2^3*3*5$ yields ratios of $\frac {8}{30};\frac{12}{60}; \frac{18}{180};\frac{16}{120} etc.$

Okay, that isn't precise but it's clear that those are the highest ratios.

$\tau(n) = 1$ only if $n = 2$; $\tau(n) = 3/4$ only if $n = 4$; $\tau(n)=2/3$ only if $n = 6$ and $\tau(n)=1/2$ only if $n =8,12$.