Is it dangerous not having air conditioning cabin filter?

Typically it's just a pollen filter that's more for occupants comfort. In older cars you would very occasionally get a dead bumble bee through a vent but I'd say that six days unfiltered shouldn't cause much of a problem for you.

Think about the long-term health of the air-conditioning system also.

If the AC is used for long enough sans cabin filter, the fan blower will get clogged up with dirt and dust much faster. This can lead to issues related to low air flow, such as lack of cooling, a frozen evaporator and a frozen expansion valve.

In itself for the short term I would say that there are no health risks

However....

Over an extended period of time there will be a buildup of organic material within the ventilation system, along with moisture and naturally occurring bacteria you could get a foul odor from your ventilation system.

As well, if your immune system is compromised nocardia bacteria can grow in the HVAC and possibly give you ill-effects regarding health in young children, the elderly and those in poor health.

I would replace the cabin filter.

There are no health risks, not all cars come with cabin air filters. The worst that will happen is you might get small leaves and crap stuck in the system depending on how big the intake is to let in fresh air.