John Tukey strongly and cogently argued for a proportion type of measurement in his book on EDA. One thing that makes proportions special and different from the classical "nominal, ordinal, interval, ratio" taxonomy is that frequently they enjoy an obvious symmetry: A proportion can be thought of as the average of a binary (0/1) indicator variable. Because it should not make any meaningful difference to recode the indicator, the data analysis should remain essentially unchanged when you re-express the proportion as its complement. Specifically, recoding $0\to 1$ and $1\to0$ changes the original proportion $p$ to $1-p$. For example, it should make no difference to talk about 60% of people voting "yes" or 40% voting "no" in a referendum; the two numbers 0.6 and 0.4 represent exactly the same thing. Thus, statistics, tests, decisions, summaries, etc., should give the same results (mutatis mutandis) regardless of which form of expression is used.
Accordingly, Tukey used re-expressions of proportions, and analyses based on those re-expressions, that are (almost) invariant under the conversion $p\longleftrightarrow 1-p$. They are of the form $f(p) \pm f(1-p)$ for various functions $f$. (Taking the minus sign is usually best because it continues to distinguish between $p$ and $1-p$: only their signs differ when re-expressed.) When scaled so that the differential change near $p=1/2$ equals $1$, he called these the "folded" values. Among them are the folded logarithm ("flog"), proportional to $\log(p) - \log(1-p)$ = $\log(p/(1-p)$ = $\text{logit}(p)$, and the folded root ("froot"), proportional to $\sqrt{p} - \sqrt{1-p}$.
A mathematical exposition of this topic is less convincing than seeing the statistics in action, so I recommend reading chapter 17 of EDA and studying the examples therein.
In sum, then, I am suggesting that the question itself is too limiting and that one should be open to possibilities that go beyond those suggested by the classical taxonomy of variables.
Addendum: Why "Interval" and "Ratio" are not quite correct answers
Stevens created the nominal-ordinal-interval-ratio typonymy in a cogently argued 1946 paper in Science (New Series, Vol. 103, No. 2684, pp 677-680). The basis for the distinctions is explicitly invariance of the "basic empirical operations" under group actions. His Table 1 describes the relationship between scale and group thus:
$$\begin{array}{ll}
\text{Scale}&\text{Mathematical Group Structure} \\
\hline\text{Nominal}&\text{Permutation Group } x^\prime = f(x);\ f(x) \text{ means any one-to-one substitution} \\
\text{Ordinal}&\text{Isotonic Group } x^\prime = f(x);\ f(x) \text{ means any monotonic increasing function} \\
\text{Interval}&\text{General Linear Group } x^\prime = ax + b \\
\text{Ratio}&\text{Similarity Group } x^\prime = ax
\end{array}$$
(This is a direct quotation, with some columns not shown.)
This must be read with some latitude, because we always have the option of choosing a model that is not exactly correct. (For example, a Normal distribution as a model of variation can be extremely useful and quite accurate even when applied to, say, the heights of people, which can never be negative even though all Normal distributions assign some probability to negative values.) Thus, for instance, data of extremely small proportions could arguably be considered as being of ratio type because the upper limit of $1$ is practically irrelevant. Data of very closely spaced proportions that approach neither of the limits $0$ or $1$ might conceivably be considered of interval type. Limiting the scope of the questions to either of these special cases would (partially) justify some of the other answers in this thread which insist that proportions are on an interval scale or ratio scale. However, when proportions in a dataset can be both large (greater than $1/2$) and small (less than $1/2$) and some of them approach $1$ or $0$, then obviously neither the general linear group nor the similarity group can apply, because they do not preserve the interval $[0,1]$. This is why Stevens' classification is incomplete and why usually it cannot be applied to proportions.
