Repeated reciprocal

Python 3, 101 bytes

lambda s:g(int(s.replace(".","")),10**s[::-1].index("."))
g=lambda a,b:a and(b%a and g(b%a,a)or b//a)

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Format: the string must contain a decimal point.

Mathematica, 36 bytes

Last@*ContinuedFraction@*Rationalize

Demo

In[1]:= f = Last@*ContinuedFraction@*Rationalize

Out[1]= Last @* ContinuedFraction @* Rationalize

In[2]:= f[0]

Out[2]= 0

In[3]:= f[0.1]

Out[3]= 10

In[4]:= f[0.2]

Out[4]= 5

In[5]:= f[0.3]

Out[5]= 3

In[6]:= f[0.4]

Out[6]= 2

In[7]:= f[0.5]

Out[7]= 2

In[8]:= f[0.6]

Out[8]= 2

In[9]:= f[0.7]

Out[9]= 3

In[10]:= f[0.8]

Out[10]= 4

In[11]:= f[0.9]

Out[11]= 9

In[12]:= f[1]

Out[12]= 1

J, 18 bytes

%@(-<.)^:(~:<.)^:_

In J, the idiom u ^: v ^:_ means "Keep applying the verb u while condition v returns true.

In our case, the ending condition is defined by the hook ~:<., which means "the floor of the number <. is not equal ~: to the number itself" -- so we'll stop when the main verb u returns an int.

u in this case is another hook -<. -- the number minus its floor -- whose return value is fed into @ the reciprocal verb %.

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