How isn't blockchain fundamentally broken? This question is a good 3 to 5 years late on the crypto trend but nevertheless I'm still having trouble to understand.
The chain works by verifying every previous block, right? How does this not become an exponential problem?
Say I want to join the network and start a node, I have to fetch the entire block chain which:

Can grow to unlimited size?
I have to verify, by checking every block of theoretically unlimited count?

I do believe the crypto currency can not grow as it does by having such fundamental flaws, but I seem to not be able to find the solutions to them.
The chain works by verifying every previous block, right? How does this not become an exponential problem?

I think you are assuming Blockchain growth is exponential with time, but it is linear with time (1MB / 10 min).

I have to verify, by checking every block of theoretically unlimited count?

In an extreme scenario, you are right. In the year 3000, we will have ~100 TBs to download and verify and it will keep on increasing ever after. But it's not as bad as it sounds like.

The rate of growth is low: Because it grows linearly with time, at a given time you will always have a finite amount of data to download and verify. As others have mentioned, there are solutions like SPV and node pruning in order to avoid full storage. If you want to run a full node with entire blockchain stored locally, this rate of increase 1MB/10min is considered small relative to the rate of growth of technology.

Increased efficiencies of node resource utilization: A lot of effort goes into managing the node resources to optimize network bandwidth, disk block compression, verification speed in order to mitigate to downfalls of ever-increasing blockchain size. In fact, if you look at the historical bitcoin-core Initial Block Download times, you will find these times have kept the same or improved over time despite the increasing blockchain size.