In combinatorics there are quite many such disproven conjectures. The most famous of them are:

1) Tait conjecture:

Any 3-vertex connected planar cubic graph is Hamiltonian

The first counterexample found has 46 vertices. The "least" counterexample known has 38 vertices.

2) Tutte conjecture:

Any bipartite cubic graph is Hamiltonian

The first counterexample found has 96 vertices. The "least" counterexample known has 54 vertices.

3) Heidetniemi conjecture

Chromatic number of tensor product of finite undirected simple graph is equal to the least of chromatic numbers of those graphs.

The first known counterexample to this conjecture has more than $4^{10000}$ vertices

4) Thom conjecture

If two finite undirected simple graphs have conjugate adjacency matrices over $\mathbb{Z}$, then they are isomorphic.

The least known counterexample pair is formed by two trees with 11 vertices.

5) Borsuk conjecture:

Every bounded subset $E$ of $\mathbb{R}^n$can be partitioned into $n+1$ sets, each of which has a smaller diameter, than $E$

In the first counterexample found $n = 1325$. In the "least" counterexample known $n = 64$.

6) Danzer-Gruenbaum conjecture:

If $A \subset \mathbb{R}^n$ and $\forall u, v, w \in A$ $(u - w, v - w) > 0,$ then $|A| \leq 2n - 1$

This statement is not true for any $n \geq 35$

7) The Boolean Pythagorean Triple Conjecture:

There exists $S \subset \mathbb{N}$, such that neither $S$, nor $\mathbb{N} \setminus S$ contain Pythagorean triples

This conjecture was disproved by M. Heule, O. Kullman and V. Marek. They proved, that there do exist such $S \subset \{n \in \mathbb{N}| n \leq k\}$, such that neither $S$, nor $\{n \in \mathbb{N}| n \leq k\} \setminus S$ contain Pythagorean triples, for all $k \leq 7824$, but not for $k = 7825$

8) Burnside conjecture:

Every finitely generated group with period n is finite

This statement is not true for any odd $n \geq 667$ (proved by Adyan and Novikov).

9) Otto Shmidt conjecture:

If all proper subgroups of a group $G$ are isomorphic to $C_p$, where $p$ is a fixed prime number, then $G$ is finite.

Alexander Olshanskii proved, that there are continuum many non-isomorphic counterexamples to this conjecture for any $p > 10^{75}$.

10) Von Neuman conjecture

Any non-amenable group has a free subgroup of rank 2

The least known finitely presented counterexample has 3 generators and 9 relators

11) Word problem conjecture:

Word problem is solvable for any finitely generated group

The "least" counterexample known has 12 generators.

12) Leinster conjecture:

Any Leinster group has even order

The least counterexample known has order 355433039577.

13) Rotman conjecture:

Automorphism groups of all finite groups not isomorphic to $C_2$ have even order

The first counterexample found has order 78125. The least counterexample has order 2187. It is the automorphism group of a group with order 729.

14) Rose conjecture:

Any nontrivial complete finite group has even order

The least counterexample known has order 788953370457.

15) Hilton conjecture

Automorphism group of a non-abelian group is non-abelian

The least counterexample known has order 64.

16)Hughes conjecture:

Suppose $G$ is a finite group and $p$ is a prime number. Then $[G : \langle\{g \in G| g^p \neq e\}\rangle] \in \{1, p, |G|\}$

The least known counterexample has order 142108547152020037174224853515625.

17)$\frac{p-1}{p^2}$-conjecture:

Suppose $p$ is a prime. Then, any finite group $G$ with more than $\frac{p-1}{p^2}|G|$ elements of order $p$ has exponent $p$.

The least counterexample known has order 142108547152020037174224853515625 and $p = 5$. It is the same group that serves counterexample to the Hughes conjecture. Note, that for $p = 2$ and $p = 3$ the statement was proved to be true.

18) Moreto conjecture:

Let $S$ be a finite simple group and $p$ the largest prime divisor of $|S|$. If $G$ is a finite group with the same number of elements of order $p$ as $S$ and $|G| = |S|$, then $G \cong S$

The first counterexample pair constructed is formed by groups of order 20160 (those groups are $A_8$ and $L_3(4)$)

19) This false statement is not a conjecture, but rather a popular mistake done by many people, who have just started learning group theory:

All elements of the commutant of any finite group are commutators

The least counterexample has order 96.

If the numbers mentioned in this text do not impress you, please, do not feel disappointed: there are complex combinatorial objects "hidden" behind them.