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Python 2, 53 52 bytes
Based on xnor's answer.
-1 byte thanks to dingledooper!

f=lambda l,t=0:-min([a&lt;t or~f(l,b)for a,b in l]+[0])

Try it online!
This is a recursive function which takes 2 arguments:

l: a list of all offered meetings
t: the time our last meeting ended (initially 0)

The list comprehension iterates over all offered meetings (a, b).
If the meeting starts before our last meeting ended (a&lt;t), a&lt;t or ... evaluates to True, which is basically the same as 1 in Python.
Otherwise f(l,b) is called to see how much more meetings we can fit in if we take the current meeting. The bitwise inverted value of this gets into the list.
If a&lt;t was false for any meeting (a, b) in the comprehension, min returns the smallest of all possible ~f(l,b) values. Then -min(...) returns -~f(l,b)==1+f(l,b) for some meeting.
Otherwise -min(...) is equal to 0.

47 bytes if we can assume that the input is not empty:

f=lambda l,t=0:1-min(a&lt;t or-f(l,b)for a,b in l)

Try it online!