Usually a creature has only a single opportunity attack available
To make an opportunity attack a creature needs to use their reaction to do so. From the rules on opportunity attacks (emphasis mine):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
And you only get your reaction back at the start of your turn. The rules on reactions state:
When you take a reaction, you can't take another one until the start of your next turn.
Therefore, you usually only have up to 1 opportunity attack available (you actually have 1 reaction available, which you can use on an opportunity attack).
Having multiple Opportunity Attacks available does not necessarily interact with the Exhaustion mechanics
There are some abilities that allow one to have more than one opportunity attack available, such as can be found in this related Q/A1. However, even in the case of a creature with one such ability that is able to make some more opportunity attacks in a single turn, this does not mean a creature will get exhausted unless the ability specifically states so.
This is because Exhaustion is a completely different mechanic that usually does not interact with opportunity attacks at all:
Some special abilities and environmental hazards, such as starvation and the long-term effects of freezing or scorching temperatures, can lead to a special condition called exhaustion.
In general, any ability or circumstance that causes a creature to gain an exhaustion level will have to state so specifically and as far as I'm aware, no ability that grants more opportunity attacks mentions exhaustion.
Differences from D&D 3.5e
You have stated that you started playing long ago so I will add a small note on how this is a deviaton from D&D 3.5e which may be confusing you:
In D&D 3.5e you had an Immediate Action available (which is similar to the 5e reaction) and Opportunity attacks were a separate thing. In 5e, an Opportunity attack uses your reaction instead of being a separate mechanic (as stated above) so there is no need for a rule that further limits the number of opportunity attacks one can make.
1. Found by illustro