What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), with an algorithm allowing to effectively compare any two ordinals written in that notation?
(I understand that every such scheme can in principle be extended by adding ad hoc symbol for $\alpha$ itself, but I am interested in notations that have been actually described.)
On
It will depend on what you mean by a "notation scheme". In general, for any computable well-ordering $\prec$ of $\mathbb{N}$ you can view each number $n$ as a notation for a particular ordinal $|n|_\prec = \{ |m|_\prec : m \prec n\}$. For any computable well-ordering $\prec$ the set of such ordinals is countable and downward closed, and is thus itself a countable ordinal. In general any order type of a computable well-ordering of $\mathbb{N}$ is called a computable ordinal.
The supremum of the computable ordinals is known as $\omega_1^{CK}$ after Church and Kleene. This is a countable limit ordinal, and there is no computable well-ordering of $\mathbb{N}$ with this order type, but for any $\alpha < \omega_1^{CK}$ there is such a computable well-ordering of $\mathbb{N}$ of order type $\alpha$.
So I would say that the answer to your question, in generality, is $\omega_1^{CK}$ if by "system of ordinal notations" you simply mean "computable well ordering of $\mathbb{N}$". Depending on how the question is read, the answer might instead be "every ordinal up to $\omega_1^{CK}$ but not $\omega_1^{CK}$ itself".
Ordinal arithmetics say that given an ordinal $\alpha>0$, and $\beta$ we can write $\beta$ uniquely as a polynomial in $\alpha$, that is: $$\beta=\sum_{i=1}^n \alpha^{\tau_i}\cdot\sigma_i$$ Where $\sigma_i<\alpha$, and $\tau_i<\tau_j$ for $j<i$ (we must have this since ordinal arithmetics are hardly commutative). When $\alpha=\omega$ this is called Cantor normal form. If you wish to consider this as the "proper" way to write up ordinals, which makes sense because then the coefficients $\sigma_i$ are finite numbers, you may want to consider $\epsilon_0$, which is a countable ordinal.
$\epsilon_0$ is the least ordinal $\alpha$ for which $\alpha=\omega^\alpha$. It can be defined as the limit of $\alpha_n$ where $\alpha_0=\omega$ and $\alpha_{n+1}=\omega^{\alpha_n}$.
Below this ordinal, you can write every ordinal in a Cantor normal form in a very nice way. Above $\epsilon_0$ you can still write every ordinal in a Cantor normal form, but you have "circular" forms because $\epsilon_0 = \omega^{\epsilon_0}$, and so it is its Cantor normal form.
Of course $\epsilon_0$ is not the only number with this property, there are uncountably many countable $\epsilon$-ordinals. In fact, one can even consider $\omega_1$ as $\epsilon_{\omega_1}$ since $\omega^{\omega_1}=\omega_1$. However it is often usual to talk about countable $\epsilon$-ordinals.
As for how big is this? Well, not very big. While this order type is nearly incomprehensible it is still only countable, thus very very small in terms of cardinality. This ordinal arises naturally in complexity proofs.