Is there a way to go beyond the tipped arrow limitations?

Is there a certain command to get some arrow such as maybe instant health 3 or more?

Absolutely, using NBT tags! Specifically, the tag CustomPotionEffects.
It takes three identifiers, id (which is the effect given), duration (in seconds), and amplifier (strength of the effect).

You can use a generator such as this one to make potions or tipped arrows for you, but I've explained how they work fully below.

Here's an example of an instant health 3 arrow.

/give @p minecraft:tipped_arrow{CustomPotionEffects:[{Id:6,Duration:0,Amplifier:2}],display:{Name:'[{"text":"Instant Health III Arrow"}]'}} 64

I'll break it down in more detail.

Id:6 is the numerical ID of the effect. Unlike block IDs, effect IDs use numbers to represent each value inside NBT, making it more difficult, so you can consult this list to find the one you want.
Duration:0 is the duration, in seconds. This tag is actually optional, and any value below/equal to 0 will be treated as 1. It caps at 2147483647 (the 32-bit integer limit).
Amplifier:2 is the level of the effect minus one; think of amplifiers as adding onto the default effect level of 1. This caps at 255, for a maximum effect level of 256.

For more information on the possible NBT values of CustomPotionEffects, you can see the minecraft wiki page here.

Note that all potions or arrows given like this will be pink, as they are otherwise unobtainable. You can change the colour with a different tag, CustomPotionColor, e.g a red regeneration 1 effect lasting 90 seconds would be:

/give @p tipped_arrow{CustomPotionColor:16711680,CustomPotionEffects:[{Id:10,Duration:90,Amplifier:0}],display:{Name:'[{"text":"Regen 1 Arrow"}]'}} 64

You'll definitely need to use the generator to find exactly the colour you want.